I ran a college anime club for a couple of months with around 200 members (I was not the president, but our president was new so he had me handle most of the operations), and I was the treasurer for over 2 years. Over the course of any given year, we had around a couple dozen events, as well as weekly anime showings. I'm no longer a member of the club since I've changed schools, so they might be doing things differently now. Our club has been around at least since the 1980s, and probably earlier, and it's changed a lot and expanded quite a bit in that time. For instance, until anime was easily available in English, they had live translations for all of their showings by students studying Japanese (which, I can attest to, is very difficult).
Here are most of the various activities we held fairly regularly. To be honest, most of this is probably not applicable for you, but it could be for others interested in starting their own clubs.
Weekly showings: Our weekly showings lasted about 3 hours, roughly 8pm-11pm. These happened every week of the semester except finals week.
- At the beginning of the semester, we voted on either one show of ~26 episodes or two of around 13 each. At the beginning of each meeting we would watch 2 episodes of that show.
- Each meeting also had a genre (e.g. mecha), and we accepted nominations for anime in that genre and voted on one of them in one of the past weeks. We would watch 4 episodes of that anime for the main showing. We tried to make genres as relevant as possible, e.g. the week of Halloween would be Horror, and Valentine's day would be romance.
- Whenever possible, we got permission from the relevant licensing organization for our showings ahead of time. In most cases this can be done via their website.
- There was also a break between the two, where we allowed random members to show videos, sold pizza for a small profit, and handled any voting and announcements.
- After the meeting, some of the members would gather in the student union and play video games or board games or just chat. This usually continued at least until 1 a.m., but sometimes as late as 6 a.m.
- During the summer, when fewer people were present (less than 20), we typically had the same meeting style but it was more open. There was no preshowing, and people were free to bring in their own collections for showings.
Fighting game tournaments: Our club made a lot of money by hosting tournaments of various fighting games like Super Smash Bros and Street Fighter, sometimes together with local gaming groups. I wasn't really a part of this, so I don't know any of the specifics other than that we had a dozen or so CRT TVs in our club's storage for this purpose.
Premier events: We often held parties when large premiers occurred in Japan, usually for big game releases. We had one for the release of Pokemon Black and White, and another for Final Fantasy XIII. The president or one of the other members would give a presentation about the history of the game series, and if any existed we'd watch a few episodes of the relevant anime. We'd also have game consoles hooked up available to play any older games in the series.
Random presentations: Occasionally our president would decide that they wanted to do something like a premier event except that there wasn't any premier going on. So we'd have a presentation about some aspect of Japanese otaku culture, e.g. visual novels, and watch some anime or a bit of a visual novel or something like that, and usually have some food.
Anime Conventions: We had a lot of people interested in anime conventions; in fact one officer's job was almost solely announcing and coordinating conventions. In order to save money we bought rooms in bulk and organized rides.
Marathons: Once a month or so, on a weekend, we'd meet for 6-12 hours and watch an entire show (13-26 episodes) in one sitting. This obviously had lower attendance, but we typically had at least a dozen or so people at these. We'd usually aim for shows that people had not watched before but were still good. Occasionally we'd also host marathons of western animation.
Holiday Parties: On various holidays or near them, we'd host parties with that theme. Halloween was by far the biggest, and we typically had at least 50 people at the Halloween parties. Many of us stayed in town during Spring break (Easter) or Thanksgiving break, or even winter break (Christmas) so we'd have parties for those two. These were not as well planned out as the Halloween party and had lower attendance. Oftentimes we'd just end up marathoning a short show with that theme. We also had parties at the end of finals week where we spent the rest of our money for the semester on food or whatever and had karaoke and video games and anime and board games and pretty much anything else people wanted to bring. All of the parties were alcohol-free and we never had any trouble enforcing that.
In conjunction with the Japanese language club, we attempted to do showings of unsubtitled (raw) Japanese anime, and we'd pause every few lines to discuss what was said. I'm not sure these were unqualified successes, since only a few members from the anime club went, but it was fun for those of us who did go.
Promotional events: My school had club fairs and cultural fairs regularly, and we'd have a table at all of them showing off our various anime memorabilia and whatnot. This was the primary way we brought in new members. We also rented out tables in the student union for the same purpose. These required quite a bit of preparation, as we needed to make sure in advance that someone trustworthy would be present at all times to make sure nothing got stolen (typically we had 3 or 4 people including at least one officer at the table at any time). We also needed to make sure we had enough business cards and people who were knowledgeable about a fairly wide range of anime to talk to anyone interested.
RPG groups: Our club was large enough that it spawned several (I know of at least 3) groups of people playing D&D and other paper and pencil RPGs. There are some anime-themed RPGs, like the Maid RPG.
Mahjong: Japanese mahjong is really nothing like the game on your computer called "mahjong". After showing either Saki or Akagi (I forget which) there was enough interest in the game to start up a weekly mahjong group. We had around 10 people who played regularly, and probably a few more who knew the rules.
Cosplay: We had a dedicated group of cosplayers who would meet roughly once a month to hold workshops and other things. I never went to one so I don't know how many people were in that group, but I'm guessing it was at least 10. We usually had a group of cosplayers at Japan night, an annual event put on by all of the Japanese-themed clubs on campus.
Collaborating with local business: One of our members started an anime store in town, and she would occasionally make announcements about new products and things. We also had membership discounts at several local stores which sold anime.
Other stuff: We were generally very flexible and open to members' suggestions and ideas. If someone had a cool idea and wanted to promote it, we'd let them make an announcement at the meetings. More than half of our events were initiated by non-officers.
Managing a club of that size is not easy, and we had over 10 officers with various roles. My guess is that you aren't aiming on expanding to a group of that size, so I'd suggest taking a few of the ideas above and adapting them to your needs.
My feeling is that for a high-school club, you should focus more on Japanese culture and less on anime specifically. Traditional Japanese culture events were mostly handled by the other Japanese culture clubs on campus (of which there were at least 3). They did things at their events like tea ceremonies, flower arrangement, Karuta, and Japanese language practice. If the people know some Japanese, you could also try Calligraphy and Poetry. You could also read some short Japanese novels. Japanese literature tends to be much shorter than western literature and is traditionally a lot darker. It can be appealing to high school students. Some examples are Kokoro, I am a Cat, and No Longer Human.
I wouldn't discount anime entirely, though. There are some very good anime which explore different aspects of Japanese culture. I suspect a lot of them will be Josei genre. Some examples might be Chihayafuru, Sazae-san, and Honey and Clover. If you're willing to include some slightly questionable series morally (PG-13 rating or so) then Aoi Bungaku, Akagi, and some other seinen works might also be a good fit. I'd also include the works of Miyazaki as culturally significant on their own, especially some of the more symbolic ones like NausicaƤ and Totoro.
Unfortunately, our website isn't very useful for anyone other than our members. However, the MIT anime club has a very good website for those interested in starting/expanding their own anime clubs. I know a couple of their members and I've often used suggestions on their page or from them. They know what they're talking about. Of particular interest are their resources page and their page about starting clubs (which has a section dedicated to high school clubs).