It is possible to attach the flex to the board without using special adhesive or tools. I have been successful 100% of the time by simply using no material between the flex conductor and the board pads. I align them up properly and use a small strip of foam on the top. Pressure is applied via the foam when the metal LCD frame is screwed into place. It's most important to clean the residual adhesive from the pads and get the flex aligned properly. I have used 3M command foam and it seems to work well.
Without special tooling you won't be able to just replace the WiFi. You can see the issue with the connection to the logic board in step 20 of the teardown guide here:
You could attempt to solder it I suppose but if it were me I would replace the logic board. Please let us know your final solution and how you do.