I would do two things in this order. First test the temp sensor for the gauge. The sensor should have high resistance (over 100k ohms) when cold but can fall to under 100 when hot. I wouldn't focus on the exact readings as much as the high to low resistance when it goes from cold to hot. Since you are looking at the gauge dropping to zero, you should have high resistance when it is hot if the sensor is failing.
If that checks out and I suspect it will, then you need to look at your dash cluster. That means you need to remove the speedometer pod from the dashboard. On the back, to pull out the cluster, you will need to release the speedometer cable from the back. The cluster may come out a little bit and then stop, try pulling gently again. It is usually the speedo cable needing to be pulled out a little and the grommet on the firewall gives resistance. Once you can get your hand behind the cluster, you will feel a tab on the metal part of the cable end, push in on the tab and the cable releases.
On the back of the cluster assembly, you should see the plastic circuit board. It will be on the outside and you will also see the dash lights being plugged through it. These are very flexible (and some would say flimsy) circuit boards. Inspect that for cracks, dirty area where the harness plugs into it.