The best I can do is share how it works on my 5210. I looked at JD parts and it looks like 5105 and 5210 is the same in this area. There is a magnetic pickup (Re519144). In front of the fuel pump. Disconnect the negative side of the battery. Take it out, make sure it is clean of metal pieces and does not appear to be damaged. Make sure the o-ring is there. It only takes 4 pounds of torque to put it back in, do not want it to touch the gear. Next check the wiring. There is 2 wires coming out the top of the sensor. Using an ohm meter test the ground. You will need to remove the cowl and disconnect the plug/wire at the back of the instrument cluster. On my tractor a green wire (325) goes from the sensor to the cluster, you may need to follow the wires back from the sensor to the cluster. With the ohm meter test the wire from the sensor to the back of the cluster.
Would not hurt to make sure the battery terminals are clean and the ground for the front wiring harness.
If you made it this far, and things are not working most likely the problem is in the instrument cluster. If you decide to replace the cluster, take a picture of the hours as the new cluster will show zero hours. Also if you do replace the cluster and your instrument glass fogs up for you. You may want to replace that, I included a URL on this.
