A twenty two rifle might well kill the fox if you are able to hit it in a vital area.
However my advice to any non gun owner would be to find a gun owner rather than purchase a weapon and attempt to kill a potentially dangerous wounded/sick animal with out first being very familiar with the weapon and the killing of animals with it.
If you live in an urban area you may also have shooting restrictions.
My advice would be to contact your state game regulatory agency and discuss this, most likely they would send out a management agent or recomend someone to remove the animal.
good luck and sorry for your troubles.