I have the same issues. In addition, I'm unable to lock the doors using the key itself. (Battery died in fob), and no A/C. I've read in other post about it being the same thing as you went through.
Where did you get yours rebuilt at, and I hope it was under 500 bucks.
Great post and great job.

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2001 X5 4.4
170,000 miles
Love our X and the way it keeps my wife and I safe and comfortable.