The Li-Ion cells inside are standardized. If you open the battery pack they can be replaced (with some skill). From my
experience the batteries tend to go out of balance which damages the one with the least charge in it.
It looks like it has 5 cells with 6.5Ah each in it. Given that they no longer work, at least one cell will be damaged. It might be possible to replace only one cell, but replacing all 6 might be more reliable (The still working cells can be used for other things, like lamps). To prevent this from happening again you will need a battery balancer (equalizes the voltages at the end of charging) or a battery management board (cuts off when the voltage of any cell gets too low). A quick check on ebay gives be 60€ for 10x 18650 3Ah cells. (Which is how many you would need to replace all).
For the same price I can get 6x 10Ah LiFePo4 cells which are a bit heavier and bigger (for the same
energy content), but safer and last longer.
If I had to repair such a device, I would remove the battery pack and build one from 6x 10Ah LiFePo4 cells instead.
EDIT: I completely forgot the important part:
A Lithium Battery pack needs a balancer and in your case also a protection circuit that cuts the power off if any cell gets discharged too much.