If the LT is plugged in and charging, it should start regardless of the charge level of the battery.
Unplug and remove battery. Press and hold power button for 5 sec. Plug in charger and see if it turns on. If so, turn off, insert battery and try again. If it again does not go on, It is possible the battery is pulling too much current. Replace battery.
If it does the same thing with only the charger, then there is an internal problem and needs serious help.
Note that this kind of problem can be economically prohibitive to repair.
You can access your data by removing the memory device (HDD or SSD), installing in an external enclosure and plugging that into another computer. Note, You may need to get more advice on accessing your data especially if you have encrypted it. It will likely object to your access in another computer and you will have to take ownership of the files in the new computer.
You don’t by change (or smart planning) have your data backed up anyplace, do you?