If you have PSE 12 or PSE 13, they should read the D5200 raw files.
If you have PSE 11, you need to update it (Help -> Updates) to get a newer version of ACR.
If you have PSE 10 or earlier, then you need to upgrade to a newer version for direct access or you can download/install the free Adobe DNG converter. Use the converter to convert your NEF files to DNG files. The DNG files can then be processed in your PSE.
If you need further help, please post in the elements forum Photoshop Elements
Also plese state which version of PSE you are using and which operating system you are running on.
The D5200 needs ACR 7.3. PSE11 or newer is required to host this plug-in. You can also use the free DNG Converter 7.3 or newer to create DNGs from the NEFs and import those DNGs into the older PSE/ACXR you might have. DNGC can be found, here: http://www.adobe.com/downloads/updates/