You can find asylums by several different methods.
The InsaneJournal Asylum Promotion [https://asylum_promo.insanejournal.com/] is often updated with entries listing and describing asylums. Asylum maintainers will join to post about their asylums to get more members. You can search through this Official Asylum's entries to find an interesting asylum.
There are at least three ways to find a particular asylum that specializes in a certain interest field. You can do a search based on an interest [https://www.insanejournal.com/interests.bml], which will list the asylums first. The InsaneJournal Asylum Quest [https://asylum_quest.insanejournal.com/] is the Official Asylum to join and ask if there's a specific asylum that you're looking for.