How do I repair wiring in 12v jump starter with inflator

Without seeing photos of the unit to get a rough idea of the damage done to the unit, the best advice I can give you is to use common sense and work with the length of the wires to guestimate where they should plug in. The battery is the easy part since positive (red) and negative (black) are usually color coded or denoted with labels. I can almost guarantee something was broken by this person.

I cannot find a service manual for this, so your best bet is to make your best judgements and see where each wire and connector lines up - your best bet is to try and find out who manufactured the unit for Westward and that may lead you to the service manual. The connectors should be designed to plug in one way and be difficult if it is done wrong, so there is not much to get wrong. However, some caution is required if such a thing isn’t present on that specific connector.

You will probably need to find compatible screws since those are likely long gone. It seems like a small SLA battery unit judging from what I seen in the photos, so if they stole the battery it shouldn’t be hard to find a replacement. If they ripped it open then you'll need to tape it closed again.

There is no need to reverse engineer the unit - you just need to replace the 12V connector. Figure out which gauge of wire you need to use, since you will probably not be able to reuse the strain relief as it typically gets destroyed. Once you find out which one it needs, order a replacement connector that’s appropriate or allows for more power if you cannot find an exact match.

After you find the correct connector, cut the old connector and notate the negative. The negative wire usually has a ridge the positive does not. Use this to determine where ground is and notate the positive to avoid getting it wrong. After making notes, cut the broken wire +1 inch off to avoid problems and solder the new one on. After soldering, seal it with heatshrink tubing using a butane lighter or torch*.

*Test the torch on something you don’t care about if you have concerns you will get it wrong. I know mine well enough to be confident with it, but you may need to do a few trial runs if you have never sealed heatshrink with a torch. I prefer a torch I know because the lighters I’ve had over the years are somewhat intrusive.