Landlords are pretty strict because there are strong laws that protect the tenant, and there are professional "non-payers" of rent that move from place to place and it can take months or years to get them out. However, if you can show a letter from your current landlord that you have never been late, show a letter from your new employer that states your salary (which will be MINIMUM) 40x the monthly rent, AND have a ton of cash to pay several months rent in advance, I bet there will be a landlord willing to make an exception.
Traveling is very inconsistant there lately, there will be a ton of jobs one minute, and then they will dry up. Also, depending on where you work, some places will dump on the travelers. I couldnt' even tell you where that would be because that is dependant on each individual unit/floor. If I were you, I would just apply to Sloan directly, I bet they will hire you with your experience, I know there are many people that come from out of state to work there. If you really want Sloan that bad, I would apply to any open positions that interest you, even if there are no peds available. Who's to say that in a year or so of working there that you can't transfer into that department, just get your foot in the door.
I am in the Tampa area now, like the area, don't love the nursing culture here. I am unsure if I want to stay at my current hospital or even in this area at all. I am starting an online masters program that I can't attend while I am in NY, so I will just have to stay here until that is finished. I havent even started yet so I still have at least 2 years to go.
Good luck with everything, feel free to PM me anytime!