Is there an error message? Or is this just a total lock-up where nothing in the DNG Converter works at all?
What is your operating system? Have you tried the DNG Converter 9.0?
As an alternative solution, why don't you just download Lightroom 5.7.1? It is free if you own Lightroom 5.2
If you updated to Lightroom 5.4 or later you wouldn't need the DNG converter. The last update for Lightroom 5 is 5.7.1, which you can get by clicking on Help->Updates in Lightroom. But simply downloading the DNG converter isn't going to make things compatible with Lightroom. The DNG converter is a standalone program that you have to run separately and create your DNG copies. Then you can import the DNG files into Lightroom. The "Convert to DNG" option is only for raw files that Lightroom already supports. There are some users who preferred to routinely and automatically convert their files to DNG.
Quote, "Any ideas? I really do not want to uninstall and reinstall Lightroom, since I'm working off my second download and would have to pay to get another copy of the software."
Your first two concerns have been addressed in earlier replies.
Just to clarify, your Lightroom license key allows you to have it installed on two computers (Mac & or PC) with one in use at any point in time. You can just uninstall from one and install on another as many times as necessary, so just update to version 5.7.1. Also you do not need to uninstall before Installing the updated version the older version will just get replaced by the new version. Every install package is the full application not a patch to the earlier version.