My friends cat nearly lost his back leg when playing too close to train tracks! The skin and tendons had to be grafted back and stitched up. He was in a large dog crate for nearly 3 months, and his 5 brothers / sisters had free roam. He was actually relieved to not have the daily hassle, and used to sit there lording it over the others who had to watch him scoff his private food in peace! Even had his own personal loo....
About ten years on, if Tanya gets the crate out for her mad dog to go into, Harvey is stood at the door wanting to get in and have peace and quiet! And at 15 years old, he's forgiven but never forgotten - To him, being crated has all good memories!
If you are worried about the heat, get a heavy white cloth to cover the top with, and use a portable fan to cool the room / circulate the air - Not pointed right at the cage, though!
You will feel guilty, because Lilly has the upper hand,
and will use it against you - But, secretly, she will be relieved! And anyway - That is your job!
Best wishes for a speedy recovery xxx