Search for the term "battery drain" on the forum. You will find countless posts linked to EVERY new release of IOS. I have commented on this literally dozens of times. Here is an example -
http://www.ipadforums.net/ipad-os/67642-5-1-killing-battery-ipad-2-a.html#post477224
There is no IOS 6.1 bug that we are aware of that:
1) causes the iPad battery to drain significantly faster than expected
2) causes any kind of WiFi connection problems
These two issues crop up regularly after people upgrade their IOS devices. The working theory is that the update process has gone wrong in some way, leaving a process on the iPad in a panic state (which means it will run flat out without stopping, thus draining the battery).
As for WiFi issues, it is assumed that a configuration setting, certificate, or other file has become corrupt and is preventing proper communication over WiFi.
These seem to be artefacts of the update process. Apple has done a fantastic job on supporting updates between versions of IOS, but they cannot possibly cater for every combination of previous IOS and settings that folks will have on their devices. The "older" the previous IOS, the more challenging it becomes and the more likely that there will be issues. Apple expect you to update to EVERY release of IOS on the device and not leave big gaps in between, but folks often do...
This is really no different to upgrading a desktop OS such as Windows or OSX. Most of the time it will work, but sometimes you need to bite the bullet and perform a clean install to resolve issues.
The resolution for this problem on the iPad is to perform a FULL restore to the latest version of IOS 6.1 and to restore your backup.
It may not be possible to restore your backup data and eliminate the problem altogether, because the problematic "setting" or file could be in your backup, but it is worth trying to restore your backup once after a fresh restore.
If the problems persists after that, restore the device again and set up as a new device when prompted in iTunes.
A cleanly restored iPad on 6.1 does not show any symptoms of battery drain AFAIK (and I have 3 iPads in the house running 6.1)