15hp Koehler, new battery, solenoid, starter, coil. While trying to start the fly wheel turns approx 1/4 turn then stops with some clicking. When I help rotate flywheel and try again it will start.

Starter soups but doesn't engage. Clicking sound: you hear it spinning with a click sound before it starts spinning. No grinding sound. The mechanism to engage the fly wheel could be faulty, probably it is seized, or battery doesn't have enough power to make the solenoid fully pull mechanism (check battery amps charge if low. You may need to look at the starter as someone turns the motor over and make sure it's engaging all the way to fly wheel.
Sometimes you may need to Shem or file down where the two bolts go into the starter too get it far enough into the starter casing to get it to engage.
Walld"

One portion of the ignition switch should send voltage to the starter solenoid, sometimes attached to the starter motor, on older cars, on the firewall. You can follow a lone cable from the battery (+) plus side to the starter solenoid/starter. It could be corroded where it connects to the solenoid.
So basically, the battery sends voltage to the dash, the ignition sends voltage to the solenoid in the start position, which activates the solenoid so it can send the high current from the battery to the starter. The solenoid has large contacts so it can carry the heavy current; that's why it's there before the actual starting voltage to the starter.

(Do the dash lights, etc light when the ignition is turned on? Does the horn blow? If not, all this is moot.)

sounds like a flat battery .did you put all wires and cables back on in the right way recharge battery and try with a trickle charger

what is your battery voltage ??? before and during start

Replace the starter solenoid. Make sure you hook the wires back properly. Try jumping across the solenoid with cables to see the starter crank. That makes a direction to the starter. That's what the solenoid does. It connects those 2 cable ends together to make the starter crank over.

Shim the starter

are you sure you got a good solenoid?souds like it's not kicking the starter bendix out for the start.if it hums that means it's turning right? it has to be the solenoid not kicking the starter bendix into the flywheel to start the engine.i would try another solenoid. thank you for choosing fixya.com

Without hearing it I would only be guessing, however I believe starter solenoid is a good guess.
When you turn the key the first position it applies power to the vehicles ignition coil and other operational components. The second position applies power to the starter solenoid which moves the starter gear against the fly wheel so the starter motor can crank the engine to start the combustion process. If the solenoid does not move the small gear against the big gear you will hear the starter motor running with no resistance producing a high pitch whine. Some of the gears could be missing on the fly wheel but it would not start unless you physically moved the vehicle forward to manually move the fly wheel.
Therefore, I would have the starter (starter and solenoid are usually together) checked or replaced. If you have some mechanical knowledge you could do it yourself with minimal tools. I believe turning the key back is not helping you. You just need to turn off the ignition and try to start it again. Sooner or later the solenoid will fail completely and not work at all. I would not want to be stranded so I would fix it soon but it could work intermittently for a while.

The most common cause of a "No Crank" condition is a bad starter solenoid. The starter solenoid is a large coil that, most often, sits on top of the starter and engages the Bendix drive with the flywheel and sends power to the starter. If the internal coil goes bad, the solenoid won't engage the Bendix or turn the starter on. This condition will not cause any noticeable noise other than the clicking.

your battery may check out fine, being 5.9 liter take a big amperage to turn that starter over 12volt does not mean a thing, amperage is what count. Every auto part store they sell many types of battery and they all 12V but different amperage. Your battery may check out 12V but amperage may have gotten weak. Check all your negative ground that you have good contact to body chassis. You can also try to get a jump from other truck (friends) that has a higher or same amperage as yours to see if it turn over. using jumper cables connect positive cable straight to your positive post and the negative directly to your motor metal ground and see that will also assist the amperage to turn over that starter. This way we can truly eliminate the battery if it does not start. If that don't work than its you ground connections or positive contact. check both positive and negative cables from your battery making sure the wires is good. Inspect the ground wire to your starter making sure it has good contact. Oil and rust will make give a bad contact. If your hear the serenoid clicking when the key is turn over and you can hear the starter making a strong clicking noise to the flywheel but not able to turn your fly wheel, and we try to jump with addition battery increasing the amperage along with good starter, all the gound wires check out, chances are your motor may be lock up. The only way to check if your motor/flywheel is lock up, take all the spark plugs out that way it will reduce your compression, take a big close end box wrench and put it on the main crank pulley and try to turn the main crank pulley over by hand. Go from there and re-post and let us know, one of us will try to continue to walk you through it.