Currently my stem is way to long for me and a little steep in angle. I've attached a photo of the kind of thing I have currently (too steep). 90mm with a less steep angle if possible .
OK to drill a good solid forged aluminium stem such as Cinelli/GB/SR/3TTT etc.
Don't do it to a modern welded tubular stem.
Plenty of touring and cyclocross bikes surviving from the 60s and 70s like this.
Drill about 30mm from the stem bolt, a 1.5mm hole through all for the cable inner followed by a 6 mm diameter, 3-4 mm deep in the top of the stem (to hold the cable outer).
Make sure it's centred and (if possible) parallel to the quill (not square with the extension) to get a clean looking cable line.
Although it would be possible with a vice and hand-held drill, probably better to do it with something more solid and accurate, like a milling machine, really.
Do you know any old blokes in sheds, model engineering, old motorbike etc?
BITD drilling the stem was the way - clean, lightweight, uncomplicated. But as old parts become rarer, you may not want to shed-modify them. Plus, if you have concealed cables, its gonna look awkward
Your current stem looks quite like an MTB one, and so the 1 1/8" column size to fir their wider diameter steerer tubes. If it's not though, then I've a 1" one for a tourer bike that might fit the bill?