Are Coaxial DC Power Plugs with Grounding available?
The power adapter to my LCD Monitor is three pronged, implying grounding


Assuming you mean a three-pin (three-pronged) AC connector on the power adapter something like this:



(Edited from this image on this web page)

... and not just a three-pin mains plug then yes, that suggests that the power adapter has a ground connection.

However that doesn't affect the type of DC power connector which is used - it just means that that the negative side of the DC output is probably ground-referenced.

Unfortunately, for this specific symptom that you stated:


  It frequently comes off and I needed to tap it in, but now even that does not seem to work.


... the problem is usually that the "socket" in the device (in your case, that's inside the monitor) has become worn / damaged / broken away from the PCB etc. and the cause is not the "plug" on the end of the cable.

This picture of an "open frame" DC power jack (i.e. socket) makes it easier to see its flat spring (marked with the red arrow):



(Edited from this image on this web page)

That spring contacts the outer surface of the plug and pushes the inner surface of the plug onto the pin. Therefore it is the socket which provides the friction to keep the plug &amp; socket "joined"; replacing the plug on the cable (as you suggest) is unlikely to help your problem, based on the "it frequently comes off" description you have given.

If you really want to replace the plug then, provided that the DC plug you have found has the same external and internal (i.e. pin) dimensions, and is visually similar (i.e. no non-standard notches, screw locks etc.) then that is what you need to know to get a replacement.  See a list of possible sizes at the Wikipedia page for Coaxial power connectors.

It can be difficult to tell the difference visually between a 2.1mm pin and a 2.5mm pin, for example. However if you are sure that you require a 5.5mm x 2.1mm plug, then the plug + cable which you linked matches that description. The plugs are also available on their own. (If you have similar connectors on other power supplies, with known pin sizes, then you could "test fit" those to your monitor with the power off i.e. no mains connection to the "test" power supply, in order to try those known pin sizes. As Jasen suggested, a 2.5mm drill bit shank can also be used to differentiate between a 2.1mm and 2.5mm hole in a plug.)