I don't know the carrier you use, and I don't know what insurance you have. I recommend pulling out your insurance terms and familiarizing yourself with the options your specific insurance offers.
I use sprint for example. The in this case, if they determine the port is physically damaged, they will most likely have me use my insurance through asurion. I pay a deductible and I get a replacement device.
If you can't turn on the phone to get any data off then you are not going to be able to backup the data. This is why it's important to use cloud storage and do routine backups on your own to your personal computer.
If you send it off to htc directly, they may be able to replace the port (if the port is indeed faulty /has failed) and retain your data. It's worth a shot to open up a query directly to htc.