Actually, no. While she may not have been serious when she said he 'slipped on the soap', that doesn't automatically mean he did die of AIDS. It could be the case that Hana just didn't want to talk about it, and given the lack of gravity in the scene, it doesn't seem likely that it would be where they reveal something like that. Furthermore, even if he did contracted AIDS and died from it, that doesn't mean he passed it on. They could have been having only safe sex.
Furthermore, since it's never revealed outright, and given the only information we get from the hospital scene — which was that the lifestyle of a homeless person is making it worse, it would make much more sense that her condition wouldn't have been anything serious if she hadn't been homeless. Especially since it would ruin the entire hopeful tone of the story, and the especially upbeat ending, if one of the characters was going to die anyway.
Bronchitis, for example, would be a very bad thing for a homeless person to get, but it would be completely non-fatal to someone able to get medical attention.
As Hana would be able to by the end, because they won the lottery.