Was she a liceensed provider (or whatever the equivalent is in CA?) I am in MN, and for a licensed home childcare provider, it is against RULE 2 (home childcare..Rule 3 is for centers) to have any child sleep in a car seat, swing or anything other than a crib. It has to do with SIDS guidelines for one thing (By the way, RULE 2...its the whole complete set of regulations....miles of them down to every last teeny thing, that we have to follow...its just called "Rule 2").
If nothing else, look to your State guidelines..even if she wasn't, it could be a frame of reference for WHY its not a good idea (safety reasons, due to SIDS info...also...car seats are not intended for sleeping!)....
Also, 60 days is insane....the norm here is 2 weeks..but personally, my contracts state a one day notice for me or the families is acceptable (why, if there is an issue in place, would either of us want the relationship to continue??) but for good situations, 2 weeks or more is nice for both parties (like a move to a different program or hours no longer work, moving homes, whatever).
Ok Another thing...definately play up the "my child was in danger" sort of aspect if you do indeed go to court. I had a fantastic contract basically forgotten by a judge when a parent ...who was in the wrong...I canned them due to 3 yr olds awful behavior after 6 mos of behavior plans, etc..I did it by the book...this was when I had a 2 week notice..pre paid to cover the last 2 weeks in case people skipped out on me. They took me to small claims court to get it back..2 kids worth....and as soon as she pulled the "my kids were in danger" card, the judge sided with her. I paid it back...got investigated by the county during this time (found to have "not occured" by the way cuz I have great documentation and pictures of the injuries the kid caused by biting and hitting others...I am great with paperwork!) And I moved on...and part of why I do the one day notice time now. Live and learn.
Good luck..let us know what happens!!