I figured out what it was. I inspected the "EventLog" table, and I sorted the results by LogCreateDate. The most recent log item had an "ExceptionHash." I copied this hash, and selected the row from the "Exceptions" table that had this same hash. The message for this exception was:

"Could not load file or assembly 'Newtonsoft.Json, Version=6.0.0.0, Culture=neutral, PublicKeyToken=30ad4fe6b2a6aeed' or one of its dependencies. The located assembly's manifest definition does not match the assembly reference. (Exception from HRESULT: 0x80131040)"

This led me to believe that the reason why the persona bar was not appearing was due to the version of Newtonsoft.Json being different than the version that DNN was expecting. I fortunately had another instance of the same version of DNN on my computer. I went into this version's "bin" directory, and I right clicked on the "Newtonsoft.Json.dll" file, and went into the "Details" view. Under "Product version," it said "7.0.1.18622." I then looked at the version in the first site's "bin" directory. This version said "8.0.3.19514." I then replaced this newer version with the older version from the other DNN installation, and I restarted my application in IIS. After refreshing the page, the persona bar started appearing again.