Heads up. That grey stuff "that looks melted" is thermal compound. It's supposed to be there. If you removed whatever was ontop (I don't know what the inside of the Wii looks like), then you need to remove and replace the compound. The compound needs to have no airpockets in it and between the two things it sits between. Even really tiny ones will cause uneven heat transference which will cause hot and cold areas and burn out the chip.
Even if it's working "now", since you messed with the compound, you need to go buy some "silver paste", clean off the old paste on both parts (wipe off with a paper towel and then clean off with isopropyl alcohol on another paper towel until you don't see any paste residue), then apply a paper-thin layer to the chip and put everything back together. IF YOU LIFT UP once you've put the two pieces together you need to start all over!!
Also, there are lots of this which can interfere with WiFi. It's simply a radio signal. Most frequent issue is someone elses WiFi. You mentioned seeing 5-6 other routers, which means that everyone's signal is broadcasting over each other. What you should do is change the channel on your router and match it with your devices. Not sure what router you have, but it's in your web-based interface and in the properties of your WiFi network card. The Wii will change automatically based on the SSID you use.
As mentioned, cordless phones can also interfere. Especially cheap ones. Those tend to use both 2.4 and 5.8 ghz frequencies.
My first guess is that someone within 300 feet just bought a new WiFi router and either it's Wireless-N or a "speedboost" G. Both of those ramps up the radio power and can drown out other routers.
Central Ohio: Wii via component on Mitsu 55" HD-RPTV. 94 ImpalaSS, MCSE/MCT, MS/Network Consultant.