Wiimote battery socket is not working.

Sounds like (maybe) two issues.

1. Bent contacts: As far as the bent metal clips are concerned, try using needle nose plyers to orient them correctly (bend them back). If this is part of the upper set, they are one piece that should be able to be removed by just a flat blade screwdriver and/ or needle nose plyers. Once removed, it might be easier to shape/bend back in place. If the lower set is bent, then you might consider removing the back case to access the metal connectors better.
2. Not powering on: If batteries can be held in place temporary to make good contact and the wii remote does not power on it may be more serious and requiring you to open up the remote to inspect.

A few hints in either case:

• Corrosion from battery leakage can be a problem and if so, would need to be cleaned out completely. Corrosion is usually visible on the upper metal plate as a dry green/white powdery substance stuck to the metal plate. Cleaning toughly with vinegar and baking soda, will help and may result in fixing a power issue.
• Bent contacts may result in batteries not making full contact. When holding vertically, the lower contacts are (-) on the Left, and (+) on the Right. The upper contacts are one plate allowing the two batteries to connect in series. As long as the contacts touch the batteries, the wii remote should get power; not easy to check the lower contacts from the battery compartment without touching the batteries, however using a voltmeter and checking around the lower contacts you should measure >=3vdc and this would confirm the upper contact as good.
• Power button has been called as a component to be consider for replacement, although the controllers I've seen have button switches that are unlikely to get dirty or need replacement by design. However, and if so, sourcing the part and taking the undertaking to replace may not be course considering time and cost. One could perhaps put some alcohol or de-oxit around the switch to see if there was any dirt or corrosion in the switch area.
• Electronics inside might be for the more astute to troubleshoot, and I am not the best to advise. There are test points on the PCB to troubleshoot further and can result in finding a faulty regulator, circuit, or IC. This would take a lot of work and know-how to get deeper into.

Suggesting: Try everything you can and consult others to see if you can fixt it. if not, pickup a replacement controller if you need it but keep the old one...Good to experiment further on the old one, or use for parts if another fails, or something.

This response is some ~6years old, but still might help others.