I think you have a 1985 Suzuki LT250EF, I could be wrong but here is a picture of the one I have. The starter for these should be on the front and I would check the starter put a jumper pack directly to the starter for a brief second just to see if it has anything left in it, if it turns then it might still be good and it could be a bad solenoid or some other electrical thing. (I don't know much about electrical stuff I kind of just replace things until it works so I would ask someone else about that) The piston rings could be a problem I would also check valve seals, this has dual over head valves so it is likely a valve seal could have gone bad, taking the valve cover off is easy on these just don't break the small bolts soak them in PB blaster or something first, then if they still feel like they are going to break heat up the metal around them. If you need anymore help I did a top end with piston rings and a cylinder hone to this engine so I can help with that. Here is a link to a parts diagram https://www.suzukipartshouse.com/oemparts/l/suz/50d08585f870022710159ef4/1985-lt250ef-parts
I have a electrical diagram for it too, I can post a picture of it if you want.
The rack I have on mine is not original neither are the rear tires. Mine has no rear suspension only front suspension, has the reverse level on the left above the foot peg and the gas tank is next to the rear tire.
I have a 12V 9AH battery in mine and that works, a 12V 12AH battery would probably fit though.