The Insignia Flex series run an x86 processor if I am correct. I have the Windows 10 version of the same terrible tablet. :| In Android they would have to provide an APK to the Play Store specifically for x86, as well as your version of Android for it to run I believe. This is why your choices are severely limited.
If you have a multimeter I would almost check the connector on the keyboard for a short circuit or a pulsing voltage coming from one of the leads. Sounds like the keyboard is doing bad things to your device. Wouldn’t advise using it at all. Try bringing it into a Microsoft store if you have one and showing them what it does (on one of the floor models.) :)
There are a few possibilities here. 1. Broken charge connector: If the connector has been handled poorly, it may no longer allow a connection between the cable and the device. This usually means soldering a new connector on to the board. They are quite tiny. You can also attempt to clean the connector with some contact cleaner, which is pretty cheap at any hardware store. Failing that, youc an use isopropyl alcohol and a tiny coffee stir-stick or other tiny implement. Wrap it with some paper towel, soak it in isopropyl alcohol and rub the connector carefully. CAREFULLY. 2. Battery has been allowed to fully discharge past its zero point: This can happen if it has been left discharged for an extended period of time. In this scenario it is completely unsafe to charge the battery. Copper shunts may have formed a short circuit within the battery. It would light on fire if charged. The battery will refuse to charge itself. (Tiny IC should prevent charging.) Those are the likely scenarios I can think of. Since...
A new keyboard is $20-$50 from what I can find on eBay just quickly Googling your laptop. Possible scenarios I have encountered in the past: (1) spill or other damage : the solution is to replace the keyboard here. Find the disassembly instructions probably here on iFixit. It should be simple. (2) battery swollen, pinching flat-flex cable : this is really more of a MacBook Pro issue but sometimes the built in lithium could swell causing the flat-flex cable to become disconnected. This could also happen due to some rough handling of the laptop or heavy handed typing flexing the keyboard itself. The solution here would also be to attempt a disassembly, then carefully check the connections for the flat-flex cable leading to the keyboard. Hope this helps! If you put any more info on here I can probably help more. Here are some links to disassembly videos. I can’t actually see a link on the Alienware iFixit page: https://www.google.ca/search?q=Alienware...-8
If you have a multimeter could you please test the test points around the board and give me a list of a voltages?
The WiFi unit on the side has two antenna ports but the Bluetooth on the front doesn’t appear to. Sorry. :|