Not sure about the BK end of this, but - I believe an eviction takes 30 days. Usually though, most properties won't even file for eviction until the rent is like two to four weeks late. So basically, while this is sneaky, what I recommend is - ask the landlord about late fees, like how late you are allowed to be and such. They will gleefully explain late charges and about eviction (landlords love to evict for some reason)

If you can include the lease in the BK, great, of course ask your attorney - how soon after filing would you have to be out of the place. Do NOT discuss the BK with the landlord at all. If you can make it thru June, just make sure you and your belongings are out before filing, and let your attorney inform the property. If you can stay there thru June without paying, I say do so, might as well save your cash. I mean the lease is just going to go into the BK anyways.