Hi,
I have the Surface Pro 1, bought the Power Cover almost 2 months ago and experienced the same issue. Tried so many things - connecting, disconnecting in 2 min. intervals, got it replaced and tried the same thing on the new one to no avail. I finally accepted the fact that it won't work as an extra battery but I kept using it as I would normally use a touch/type cover (kept it connected most of the time, used in desktop mode, tablet mode , etc.). Now, finally after 2 months (actually just now- it's status in "8% and charging"), it has started taking charge. I have read many posts where they said they kept using it and it finally started taking charge after some time - for some, it happened after a week and for others, it even took a month or a couple of months (in my case). So, you might not wanna try the surgical alternative (as explained in other comments) on your Power Cover unless you desperately need it to work soon. If you can wait, just forget that it can take charge and keep using it normally, and possibly, it'll start working sometime. It could not be a mere coincidence that it started taking charge as there are so many posts and users which/who had the same experience. However, the time varies in each case which could possibly depend upon the status for each Power Cover.