Replace keyboard and install new keyboard

You need to get it from someone with a good warranty since it’s a known problem on some of the Haswell machines. I go through PartsPeople for troublesome parts like this since they provide a 1 year warranty. Sometimes you can find used ones, but the issue is that you are more likely to have problems again, especially if it’s a known issue.

On my E6540, I went through the factory keyboard and ~4 replacements (3 being warranty swaps) so far. My current keyboard is a PartsPeople purchase and has held for 6 months and counting - the warranty replacements from Dell were far worse.

Even though the 5440 isn’t like the 6000 series where you sometimes see it more often then other machines (the heat probably doesn’t help on i7 spec systems), you may still want to go through a vendor with a good warranty to cover for when it fails again. You can find the keyboard here. The backlit keyboard can be installed as there is a controller board as an upgrade (or if that’s all you can find), which Dell did on purpose to make them easily interchanged during assembly and makes it easy to upgrade.

The video for installation is in @mayer’s answer.