All right so from what I gather from the information you so kindly provided is that you have a broken wireless adapter for your xbox 360 with a power problem. If you have a warranty i highly suggest you go about invoking it and getting a new one, as the steps I'm about to explain can take about 30 minutes to an hour of intensive labor on your part in order to get your device working again.
If you do not have a warrantly let me run you through what we're going to do really quick and what you're going to need. You're going to need a pair of needle nose pliers, the type with the grooves on the inside, not the type with a smooth one. This will be far your easiest option, but failing to have this a flathead screwdriver will probably do. In addition to the needlenose pliers you're going to need a sealent of some sort, superglue, epoxy, Ducktape.
Personally for me the most professional looking solution was to convert my white adapter into a "black" one using electrical tape and superglue, although it truly comes down to your own personal preference and what you want the end result to look like. With the needlenose pliers we're going to cut open the plastic casing (Just trust me on this one) and remove the front cover with the light.
Now this is not going to be a very nice looking solution but it's going to get it to work. Taking the needle nose pliers you should open them up and run the grooves along the line where the front of the case meets the back of the case. This is actually glued together so there is no other option that doesn't put your electronics at risk. Do this continually until you have a hole then using the holes as a weak point, expand it using longer swipes. Once the top half of the area where the covers meet is gone take the needles nose pliers, insert them into the hole and pull the pliers apart. This will separate the covers allowing you to look at the guts of the device.
There's three possible problems with your adapter. The first is impossible to fix without spending as much on a new adapter, and that is if the powerchip or brain of the device has been shattered. If this is the case let me tell you some sound advice, when asking for help provide as much information as possible, don't leave out the obvious part. But knowing that such is probably not the case, let me describe the other two. The first repairable problem is the most common when there looks to be no outright damage. And the problem is just that the power cable has come loose from the powerchip.It just manages to wiggle it's way out and then BAM! You have to buy a new one. Unless you take it upon yourself, as you have, to fix it.
So if this is the case simply plug it back into the only socket on the power chip, and put the casing together your preferred way. I highly suggest using superglue and electrical tape as it provides a good aesthetic look and is extremely functional. Now if the aforementioned plug is not disconnected there is only one other problem possible, and that is something is wrong with the cord itself. To save yourself a lot of headache cutting out the bad section of a cord i suggest buying a junked adapter on ebay for four dollars after shipping and taking it's cord, but if you want to do everything on your own I highly suggest doing this. Nothing.
If you need me to tell you what to do for the next step it has a high probability of going wrong. Electrical wiring is very finicky and if patched wrong could provide a fire hazard. Trust me and just buy a damaged adapter and take it's power cord. If there's anything else i can help with just ask and I'll do my best to help. Good luck and enjoy your now unbroken wireless adapter.