I have a Huskee log splitter with a 12.5 Briggs and Stratton engine that kicks back so violently that it broke my hand. I took it to two B and S dealers and they could not find anything wrong with it, but it still kicks back. I decided to change from a pull start to an electric start. I replaced the fly wheel with an older fly wheel that had a ring gear and put an electric starter on. My question is what size battery do I need and do I need to put a solenoid starter or can I just use a heavy duty switch to run the starter.

200 cca (cold cranking amps) at the least... I'd use around 250 or more cca.
The starter... you could just run a switch (to temporarily give power to the starter to turn it over to start) but don't leave it on... and some way to kill the engine... simplest thing would be to find a old riding mower with the wiring intact... and scavenge the wiring off it...