Hooray, I fixed it.
After fumbling around with ndiswrapper and the included bcm43xx-driver, it all just seemed pretty messed up. So I just downloaded the Dapper iso image, and installed Dapper Drake from scratch.
The first thing I did was to blacklist the bcm43xx-driver which is included in Dapper Drake. How to blacklist it is explained
here in the first paragraph.
Thereafter I downloaded ndiswrapper 1.17. I specifically did not install it with the package manager, as that gave me some problems before. I followed the instructions
here in detail, and after I installed the drivers, my card didn't show up as disabled or unclaimed anymore. A huge progress.
You may experience the following issues during the process:
1. When running "make", it didn't find the kernel build files, so I had to install linux-kernel-headers from the package manager.
2. I wasn't able to connect to the Internet to start with, but a reboot did the trick.