There are no references to KC not having a place for Caylee to stay on the nights of June 15 or 16. According to AL's statement to police, there were several instances in late May and early June when KC either couldn't go out or had to return home because of Caylee.
One day during the last week of May (exact date undetermined), AL wanted KC to "hang out" with him and another couple (his friend Clint and Clint's gf). KC couldn't leave home because of Caylee. She told him that she was at "the nanny's" and "Caylee had awakened and wouldn't go back to sleep.
May 31st, AL played music at a pool party. KC attended. The two later took in a movie. AL wanted her to stay over, but she left in the "early morning hours" after a phone call from CA.
June 1st, the two attend a party at the Pegasus Apartments. KC doesn't spend the night.
Another night that week (established by investigators to be June 4th), AL and KC attend a party at Voyage Nightclub with friends, afterwhich the group returns to AL's apartment. Casey had to leave because, according to AL, "her mother wasn't allowing her to stay over".
Three weeks after they first "spent time together", the tart mom finally spends the night of June 13th at AL's apartment. That was a Friday and CA would've been off the next day. Maybe Caylee did spend that night with her gp's. I don't know. The next day, KC told AL's roommate Cameron that Caylee was with a friend at Cocoa Beach. She later told him that she and her mother were "always getting at each other's throats".
These statements can be found in the individual interviews and in the Supplemental Report released November 26th, otherwise known as the 2nd doc dump. I don't have direct links because I downloaded them to my computer. You can find the links in this thread:
http://www.websleuths.com/forums/showthread.php?t=71772
While we can't be certain she didn't have a babysitter for June 16th, we do know Caylee was cramping her lifestyle in the weeks prior.