Aspirin doesn't dissolve blood clots. It's used more to prevent them. And it doesn't cause ulceration either but that is a good reminder of why you should always take tablets with food unless specified to take on an empty stomach.
To dissolve clots we normally use warfarin (Coumadin usually, or marevan occasionally) or enoxaparin AKA clexane injections which are injected into your fat layer with a small needle.
Both stay in your system for several days.
King will be fine. I believe the clot is in his hamstring muscle, not his veinous system, so most likely very minimal risk.