Sure, adding a drain to a lower level should help if the elevations are correct and the pipe is large enough.

There should already be a pipe running away to a lower level and dissipating the water into the yard; if you have water feeding in there, it's probably connected to a drain at a higher point in the yard, and running down to the window.

Not a good situation, one that needs to be addressed.

You may need to consult an expert, unless you see yourself digging the current drains up and inspecting them.

I suspect what has happened is the existing system is blocked with silt or tree roots, or both.

Just thought of another point, where do your eavestrough/gutters drain to? If the water just pours onto the ground, it's probably contributing to that.

A kind of half baked approach might be in the meantime to put a sump pump there, since you're risking or may already have all sorts of damage because of this.

I have been thinking of putting such a window in like this, I am in a dry area I guess one of those plastic covers for the lightwell wouldn't help, since