The controller case if you have the broken bits can be repaired by suing a fibreglass repair kit using thin GSM matting material kits, fibreglass is really simple to use if you plan what you are doing.
Cut the glass matt to size required, a braid the plastic it is to go on to with 60 grit ali oxide sand paper. Pour a small amount of resin 50grms in to a pot and add 0.5 - 1ml max of catalyst (1 - 2%) stir thoroughly then paint the mix on to the plastic add the matting and then dob more mixed resin on top so that it is soaked through and thoroughly wet. Set it a side and 25- 30 mins later it will have gone off. You can then rub it down to paint or add another layer.
The speed sensor is replaceable if you can solder, either cut the wires and solder the new sensor wires to the existing ones in the right order or better still undo the controller base and solder the new sensor wires direct to the existing pcb.
The fact the battery lights up when plugged in is a good sign but obviously when unplugged looks like the BMS switches it off so I suspect unbalanced cells and one lower then 3v.
The hard bit is balancing the battery if it is retrievable, it need sot be opened and manually inspected first to measure the cell voltages.
All the jobs are a bit fiddly and will take a bit of time but if you are not very good at diy or fixing things then it is all but junk unless you know someone near by who can help out.
Where do you live ?
Any where near West Sx ?