Temperature gauge not working

I'm currently going through all the niggling issues I've had on my Caddy since I purchased it and one of these is that the temperature gauge on the dash doesn't work and never has as long as I've owned the car. Its been off the road so haven't prioritised this much although now I'm driving it I'm trying to get this fixed. Everything else on my dash works which makes me think it's not the gauge itself although I'm not the most mechanically minded so could it be just the gauge?

My immediate thought was the temperature sender for the gauge, although looking at other posts here, I'm struggling to determine 1. which sensor this is and 2. how to get it off.
Most posts seem to refer to one sensor being black and one being blue yet both of mine are black (shown in the pictures).

The sensor in the first picture is the one on the top of the pipe going into the engine block, the second is the one on the side of this pipe. The second sensor appears to be orange on the inside and the metal clip also doesn't seem to be fitted properly.

I hope this is enough information and thanks in advance for any help with this!
 

Take the black sender on the bottom picture the connector out.  

Using a 9V battery connect the positive side to the yellow /red wire and the negative side of the battery to the brown.  Does your meter move? yes/no.
Yes good meter No bad sender or wire to the gauge.

Do the same for the other sender connector.
if the Gauge doesn't move with either then you have a bad gauge or wire to it.

Since it is a latter Caddy you didn't mention the year but look at the Bentley or Haynes wiring diagram and it will tell you the correct wire color for the gauge.

You can usually repair the gauge by removing it from the cluster, and cleaning the bad solder joints on it.  I have a how to on that for the cabriolets but it is the same for most water temp gauges.  You can also test the gauge by using the 9V battery trick to the two posts on the gauge, and you will want to hold it there for a bit to verify that your gauge is working and doesn't return to zero.

Repairing your Flaky Water Temp Gauge | Volkswagen Owners Club Forum

Testing Digifant Water Gauge | Volkswagen Owners Club Forum

Huge thanks for all of this!

So I tried the 9v battery across both senders and neither moved the gauge at all. I've then taken the gauge itself out and tested it with the 9v battery and its working okay.
Assuming this is either the sender, thermostat or a bad wire as you mentioned.

When I took the dash out however, the circuit board on the back looks to be in a poor state, not sure if this could be the cause of the issue? I've attached a picture.

And apologies, my caddy is a J reg, 1992.

Thanks again!

Well you can loosen the nuts on the gauges, using steel wool, you can polish the posts of them and lightly sand the mating sides of the nuts.  Use a Pencil eraser on the mylar to clean it, A NEW PENCIL ERASER.

See my fix for plumping up the main connector it is in the archives how-to's as well.

Then I would say it probably isn't your sender as you took the connector off and used the battery to test the wiring.    So you are looking at a wiring issue I am sorry to tell you what you probably already guessed but didn't want to face up to…. .  

I would ring the wires with a dvom on test mode and see which one of the two it is…. (probably the colored one and not the brown one.)…

I have a somewhat similar cluster hanging around should you need any parts.