Solder or Adhesive for Wifi Module
You definitely should not solder that miniature antenna connector.

It is actually designed to be self holding.

The manufacturer appears to have used some RTV silicone to further secure it, but it's very important to note that this would not be hardware store RTV!!!.  Hardware store RTV releases acetic acid which would corrode the circuit components, an alternate formulation not intended for consumer use is used inside of electronic assemblies.

We can't know the actual situation of the installation, but for many just plugging it in would be fine.  Or you could consider some kind of tape.  Even modern TV's tend to get fairly hot inside, so whatever you use would have to stand up to that.  And don't be "that guy" who uses hot glue on this.

Exercise a lot of caution during this project; even though the CRT is gone, modern TVs still often have fairly hazardous voltages present in the power supply, for example for the backlight.