The "hole in the wall" is simply passive ventilation. Air just flows naturally, but sometimes the wrong way ........... i.e. draughts.
Mechanical extraction is "active", and much more efficient in clearing water vapour.
Assuming your square hole is at least 100mm wide, fitting a standard fan is a simple DIY job.
To avoid much bashing and cementing, make up something known as a "pattress". Just a piece of MDF big enough to cover the existing hole.
Drill a 100mm hole in the MDF, and fix the fan to it. Fix the pattress to the wall, and you're done.
Done mechanically, but you have to wire it up.
Because it's in a kitchen, it has to be done by a registered electrician. You could do it yourself, but you'd have to fill out a form and get Local Building Control (Building Inspector) in to witness the work........... really not worth the trouble.
A registered (known as a Part P) electrician can avoid all that faff.
You'll also need an isolating switch as well, but that can go at the top of the wall out of the way. Have a fan with a little pull-cord if you like, but you must have the isolator.
Extraction is rather like insulation or cheese.......... you can never have too much of it. ;o)