This is odd. I have a half-dozen old Dell laptops all running Linux and all requiring the Broadcom fixes on a pristine install. However, after the initial monkey business, kernel updates don't break my modules and they all work fine with each kernel update.
Since this is a relatively new install, back up any critical data and then do:
sudo apt-get update
The above is designed to clear out any cruft remaining from your previous install attempts.
sudo apt-get autoremove
...which should pull down the Dynamic Kernel Module Support system. DKMS allows the system to autobuild driver modules every time the kernel changes in the course of a regular update. However, if you re-install the system from scratch, you bypass DKMS altogether and it cannot intervene.
sudo apt-get install fakeroot dkms
If DKMS is not already installed, then your problem may be as simple as the fact that DKMS is absent. If it is already installed, then your problem is more complicated and will need more detective work.
DKMS is not smart enough to look back in history and recognize your previous Broadcom travails. Therefore, your next kernel update may require you to go through the whole Broadcom nonsense again. However, DKMS should thereafter automatically compile the proper module of any future kernel updates.
Let us know how it goes.