I assume that you can make it work somehow, because I've seen K20s swapped into CRXs and 88-91 Civics. I've never actually seen or heard of a K-swap in a DA Integra though, so I'm not completely sure.
What I would do is call Hasport (602-470-0065). They don't take e-mails, you have to call. I don't see a K-series mount kit for the 90-93 Integra listed anyway on their site, but I know for a fact they have other products that aren't listed on the site, so it's definitely worth a try. If not, they might be able to point you in the right direction or at least explain why the swap won't work.
Off the top of my head, you're gonna need a mount kit, axles, a cable-to-hydro tranny conversion, etc.