You pretty well have the solution figured out. There is a clog in the drain and that little toy snake is totally incapable of clearing it. However, what is needed is a plumber with a real auger to clear it. These are seldom, if ever, available to a DIY. They are extremely dangerous in the hands of a novice. Dangerous to the possibility of loss of fingers, hands, or even arms, and you can easily cause serious damage to the drain. While you didn't say so, I am guessing that before the leaking occurred, you attempted plunging the toilet and this blew the wax ring. Just a backed up drain will not damage the ring. Time to call for professional help. Most users of this forum recommend not calling companies that have "Rooter" as part of their name. Some may be OK, but many not. But whatever you do, forget trying a DIY snake.