Chances are either the valve seal has disintegrated and clogged the line, or other sediment has broken free and done the same.
Disassembling a faucet is usually fairly simple. Remove the set screw retaining the handle, then start removing the exposed nut, washers, and valve components. Lay them out in order for correct reassembly. Check for damage and debris.
If your valves are not soldered in (if they're installed using compression fittings), replacement is fairly simple. It's a good opportunity for even an occasional DIYer to get his hands dirty. Post a photo or accurate description for more assistance.
How can I get it to close fully?
The plastic handle is expendable. It will break long before the valve spindle. This is good. I would use an indelible pen (sharpie) to mark the maximum angle it turns, then wiggle it back and forth vigorously a few hundred times and see if it is improving. If so - continue, if not - stop.
The black tap handle is actually removable, it just pulls of, and reveals that it is just a glorified hex-key. Could I just use a longer hex key to provide better leverage
I would try that. but I wouldn't extend the leverage very far.
or is there a problem in applying too much force?
Most ball valves seem to be pretty robust but I'm sure you could destroy one (or the surrounding pipework) with a 6-foot lever.
Should I also grip the pipes around to ensure they aren't put under force?
Yes. I would try to do that when you start to apply greater forces than normal. Maybe it is possible to firmly clamp the valve body to a strong length of wood or steel and hold that when applying leverage to the valve handle (or hex key)
tap gently with a hammer as I turn the tap
Worth trying even if there is a 1:100 chance of having any effect.
use spray oil (e.g. WD40) and let that soak in
I can't imagine that would help, but again, it is a cheap and easy option to try, and less likely to cause damage than the other ideas you listed. So if you have lots of time, this is one to try first.
![enter image description here](https://i.stack.imgur.com/Vkfno.jpg)
If all fails, I would rent or buy a pipe-freezer and just replace the valve. You need a plan B immediately ready to hand in case this goes badly wrong. My plan B would be to turn off the water to the building using the valve in the street but that option might not be available to you.
Best Answer
If a valve has directionality, it will almost universally be cast as an arrow in of the body, as seen in this globe (aka stop) valve:
I've never seen a ball valve that is directional. If you look inside, you'll see the guts of it are symmetrical.
I would say the general rules of thumb for this are: