How can power lost after drilling into
stud be restored? Is it costly?
Do I need an electrician to restore the power to the bedroom on the other side of a wall I drilled into
studs
Related Solutions
As Legion600 suggested, you probably have steel studs as furring strips. You need to be sure you are not hitting protection plates installed on wood studs.
These protect electrical or plumbing lines close to the stud face, and you must never attempt to drill through these. Usually, if you drill several inches above or below such plates, you will hit wood. If you are hitting metal along the entire stud length, it is not a protection plate, but a metal stud. Metal studs in this application should be fairly light gauge and easy to drill through. Protection plates are about 16 GA and are not easy to drill through.
You may anchor objects to metal studs. They are sturdy enough when finish sheathing is attached along their length. Do not use toggles, the hole required is too large. It's best to use a number of self-drilling screws that are barely long enough to engage the stud face material after passing through the mount and wall finish. This avoids damaging any lines routed through the center holes of the stud. You can also use self tapping screws into pre-drilled holes, but drill very carefully so as not to damage any underlying lines when the drill breaks through the face material, and again, use appropriate length screws.
The DeWalt nut drivers (and others also) are very shallow (only meant for driving sheet metal screws (and are magnetic).
As suggested, regular sockets with a rachet handle will cover the hex head and not be pushed-off by the cap head.
You can get a socket adapter for your impact driver and then use your regular sockets: 
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Best Answer
It sounds like you drilled into an electrical wire, which will need to be replaced. The amount of difficulty involved in repairing it (and the cost) is mainly going to depend on where the wire is and how easy it is to access. For example, if it was between two outlets that are accessible from a basement or attic, it should be fairly straight-forward. If it was a feeder line running through an exterior wall, probably less so. Fixing it will basically require figuring out what is connected to each end of the wire and running replacement wire.