Hello  , To answer your question , yes , ankle sprains can cause 
knee pain. The joints of your lower 
limbs , upto the spine have a chain reaction , wherein ,  the healthy joint over compensates for the injuries occurred to another joint , proximally or distally, to keep your body balance as much as possible. 
You mention about your knee being hot and swells on putting a brace. Are you very sure that there wasn't any 
knee injury of any sort , when you fel down  ( twisting , direct blow etc ) ? 
Had there been no increase in temperature and knee swelling , I would have just said its  excess strain on the knee ,  due to the ankle sprain however swelling and increased temperature suggests some sort of  active process going on inside.- again , this needs to be physically examined and confirmed before we can reach to any conclusion.   
I wouldn't advice you to put on a brace  unless you've got the knee evaluated by an orthopaedic. Braces are meant to support the knee , but can cause space constriction around the joint too in case of an injured , inflamed joint . 
You should get 
physiotherapy for your ankle sprain- ankle sprains recur sooner of later , depending upon what activity you do , if there has not been any sort of rehabilitative exercises done following ankle sprains. Physiotherapy treatment ( in various forms like taping, 
electrotherapy and exercises )  for ankle sprains can begin  right from the acute phase , when there is significant swelling, to the chronic stages , when pain is felt only on  loading .   Also you should have your knee evaluated once again from an orthopaedic. Tc .