First off, there was a recall on those mowers due to leaking gas tank. For that check check on here. Now lets assume this got fixed;-) You need at least three things for the mower to start. Fuel, air, and a spark. Check for gas, clogged filters and collapsed lines. Check your carburetor, make sure your float is working right and that your needle valve is clean. Check all the lines for leaks and proper connection. If you are sure you do not have spark, then disconnect the Shut Off Wire at the Control Bracket and then check for spark at the Plug. If there is still no spark at the Plug, disconnect the Shut Off Wire at the Coil and check for spark. If you should still not have a spark, it is possible that your coil/or the ignition module is Bad. You can check and see that the Coil/Ign Module to flywheel gap is .020in. If you get spark at the Plug anytime during these checks, then a Safety Switch or Wire May be the Problem. The engine will turn over with a bad safety switch but will not fire. There should also be a blade safety switch, that you do not want to forget to check as well as the seat safety switch. To really isolate what is going on, I would disconnect the safety switches, bridge the wires. That way, you should be left with just the ignition circuit. If you now get spark, you know that it is one of the safety switches. reconnect one at a time and see which one causes the failure. Replace the safety switch. DO NOT OPERATE THE EQUIPMENT WITHOUT THE SAFETY FEATURES. Check the electrical diagram and see if that will help you further along. Good Luck and let us know how it is going.
Try replacing the battery terminal wires on both the ground and the positive of the battery (black and red wires) with new. Current is not always able to flow through old, deteriorated or frayed wiring. That is the only link in the chain so far that you have not replaced.
After that you may try cleaning your battery posts and terminals with a "post and terminal" cleaning tool.
Let us know if that works or if it is a different problem.