My iPhone 6 Plus won't turn on. How can I extract data?

Hello,

I have an iPhone 6 Plus. It's almost 2 years old, and it's been suffering from "Touch Disease" for the last 4 months or so. 3 days ago, the screen turned red with vertical white lines, and then stopped functioning altogether. I could no longer turn on the phone. Every time I tried to charge the phone, it seemed as though nothing was happening. I opened it up and checked to see if the battery got disconnected, but it didn't. The screen itself wouldn't turn on either, and all the ribbon cables were connected. I tried connecting it to iTunes, but iTunes wouldn't detect it either. When I pressed the home and power buttons together, iTunes would sometimes pick it up, and Windows would make the sound it usually makes when external drives are plugged in, but the phone isn't actually accessible by the computer, or iTunes, and the computer doesn't recognize that something is plugged into the USB port. I checked device manager. iTunes partially sees the phone, and says that it needs to be restored or updated. It doesn't recognize it as my phone though, it just shows the serial number. I clicked update, but then errors 4005 and 4013 show up every now and then, and then it mentions that the phone can't be updated.

I took it to a repair shop, and we tested multiple screens and batteries on it, but it still wouldn't turn on. The people at the shop said that the motherboard likely burned out from Touch Disease.

I haven't backed it up in about 2 months, and there is a lot of data on there that I would need, including contacts, text messages, and notes. I don't think my phone automatically syncs to iCloud over the air, but I can't remember, and I haven't done a computer or iCloud backup via iTunes in about 2 months. Is there any chance that my phone automatically synced contacts, text messages, and notes since 2 months ago over the air? If not, is there any way for me to remove the hardware from my iPhone and manually extract the data using any sort of device?

I appreciate any help. Thank you so much in advance!