There's a good chance the loose membrane is to blame and affixing it in place would solve the issue. Before continuing, you can verify if the membrane is the issue or it's something else by powering the laptop on with the key and membrane removed, opening some kind of document on the computer, then using a toothpick or other small (non-sharp) object to push down the center location. If that regularly causes an
e character (or whatever key is at issue) to be input in to the document, the keyboard electronics are in good shape.
With that done, try holding the membrane down (by its edges) in place over the switch location, while using your fingertip to depress the membrane in the center. You should see
e characters being emitted as before. This is the location you want to affix the membrane.
Additionally, double-check that the scissor-spring part of the key is in good shape. Grit or broken plastic in that part would also explain the key feeling off, so verify that it's in good shape before proceeding. (If it's not obvious, remove a working, known-good key and compare them.)
Putting the Membrane in Place
You might not love this suggestion because of its permanence, but I think your best bet is using cyanoacrylate (super glue). I can't absolutely guarantee it won't harm the membrane, but I think it's very unlikely.
- Power off the laptop.
- Clean the area that you'll affix the membrane to: isopropyl (rubbing) alcohol or acetone on a Q-tip should do the trick. You likely also want to clean the rubber membrane.
- Run a small bead of cyanoacrylate around the outer edge of the membrane.
- Carefully place the membrane in the centered location you found above when testing the keyboard. Gently hold in place for a minute or so until the super glue has set.
- Wait for it to fully cure. Overnight should be enough, a day should be absolutely safe. This might be overkill, but it would really suck to bump it out of alignment by rushing this part.
- Power the laptop on, test it out by gently depressing on center of membrane: it should register keystrokes like before when you tested the keyboard.
- Reassemble the key, you're done!
The upside to the above approach is that it (should) result in a permanent fix. But that also makes it feel risky, because if something goes wrong, it's extremely difficult to undo. A modified approach would be to put the membrane down in place without cyanoacrylate on it, then carefully put a bead around the edge, touching the outer edge of membrane and the metal where you're affixing it. If it turns out things went bad, acetone (nail polish remover) can break down cyanoacrylate, so you could carefully undo the process with some Q-tips and patience. But I think this approach would be less strong and easier to mess up, even if it has the upside of being (semi-)reversible.
Also, be careful when you're using the cyanoacrylate that it doesn't drip down off the metal (I think it's metal, not 100% sure, whatever the cross bar the membrane/switch sits on) and into the area below. Keep some acetone and Q-tips handy in case you need to clean up any accidents quickly.
Finally, I can't guarantee this will work and won't have an impact on the membrane -- I just don't know the exact chemical interaction. But I'm pretty sure you'll be okay and solidly affixing the membrane in place is the best chance of solving the issues you're seeing. But, you know, you follow instructions at your own risk, etc, etc.