This is the problem with more capacity or speed. You have 802.11BGN

With 802.11BG, your AP will have 22MHz bandwidth channel. Therefore you have 14 channels (USA has only 11 channels due to smaller frequency allocation). However, there are only 3 non-overlapping channels
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The 802.11N is using 40MHz bandwidth for it max speed. Therefore it actually can have 1 channel. This is why manufacturer concentrate more the 802.11n for the 5GHz wireless which has wider band than the 2.4GHz. The problem is many devices only operates in 2.4GHz band only.

If your neighbor is using 802.11N also, you cannot avoid interference with your channel. Yes, your signal is stronger but no way you can avoid interference. With interference, you have more error during transmission, thus your AP might switch to smaller band and lower speed and keep gong down if more error encountered at threshold setting.

I believe this is you main problem. You can use your phone and install apps that can scan the spectrum to confirm this. The best perhaps move to 5GHz band where there is less interference. I believe many laptop nowadays support this but not the lower end mobile devices like smartphone, unfortunately

Maybe can shield the house from external interference

Did you check if there is another strong WiFi signal(s) colliding with your channel? You need to do frequency scanning