I think I hyper extended my elbow. I need to quickly recover. How should I treat it? How long should it take to be healed?

Time. If it is just a sprain, it can take 2-8 weeks depending on the severity and treatment. Rest, ice, compression and elevation (rice) is a good place to start, to self treatment. If, however, you have any bony tenderness, restricted motion, numbness or loss of sensation, it may be more than just a sprain and should be evaluated by physician.
Gentle rom exercises. Xray ortho exam if pain or problems persists healing occurs at its own pace.