The clicking on the left is the fuel shut-off solenoid most likely. I doubt that hosing it off caused any real damage, I do the same thing and have been doing so for years.
Something is wrong with my MTD website and I can't access your mower right now but if you can tell me how the blades engage? Do you have an electric pto switch on the dash or a lever that engages the belt drive?
I would probably suspect a safety switch at first. Either the one for the clutch/brake or the deck. Check that the one on the brake is fully engaging and the connections are clean and making good contact. Do the same for the deck if you have the lever which engages a belt. Sometimes you can pull backwards on that lever when trying to start the engine and it will engage the safety switch. You can check these switches with a continuity light if you have one. Some you can simply unplug and try starting the mower to check them. You should also remove the key switch connector if you can get to it and give it a good cleaning.
You should also have a fuse that will be somewhere around the keyswitch wiring harness. I'm not sure on yours if the fuel shut-off solenoid is on that same circuit or not but it wouldn't hurt to check that fuse before checking anything else.
If you can check these things and let me know what you get, we'll go from there. Thanks, PK