How to convert, if possible a headphone from unbalanced to balanced?

Aaaaaah! If it has TRRS vs TRS like I thought on the HP side, then there is hope. If you have a TRRS cable with 4 conductors, you can check continuity between the poles when it's plugged into the headphones. If there's no continuity between any of them, then you're likely good to go.
 
OK, I got back the original cable, one end is single ended (quarter inch jack) and at the headphone end it's 2.5mm 4 pole jack and the second ring and the sleeve are tied together in the cable. I will make as soon as possible a custom cable (a 4 pole 2.5mm jack just arrived from aliexpress, the special version, with twist-lock). I don't have in hand a second 2.5mm jack, but i will make the other end two female 3.5mm jacks, split, to fit the hifiman cable. I'm aware of knauf1919 statement that the quality of the plating is poor...so I don't want to desolder and resolder the only jack I have now in hand.
If this works and I can use the HD560S in native ballanced mode, my only wish is to try if the Beyerdynamics (no detachable cable) cand be used the way I wanted....but my main goal was for more power on the HD560S, Beyers can be run also SE....
EDIT: I checked again the "China" cable I currently use and this also has the second ring and sleeve tied together, my mistake, I thought the R2 was free, not connected.
 
I'm looking at converting my old AKG240MkIIs to balanced in the following way. The headphone socket is a 3 pin Tiny XLR locking connector to 2.5mm jack. As you'd expect shared ground and L+R conductors, so:
a) replace the headphone cable socket with a 4 pin Tiny XLR male insert, thus the same socket but 4 pins.
b) take off the side plate (no need to open the cans themselves) and solder the left and right drivers on to the new XLR insert.
c) get a new tiny XLR plug, making a Tiny XLR to Tiny XLR - using AKG's balanced pin outs.
d) put a panel tiny XLR into amp
e) have the shields for the cable connect to the amp chassis ground through the metal of the plug itself.

The plugs/parts I would look at are switchcraft: https://www.switchcraft.com/assets/1/24/TB_M_SERIES_CD.PDF?4797

I think point 'e' seems the best idea rather than attempt to use a 5 pin and bring any noise closer to the pins of the amp.

With the cable - I don't have any idea of capacitance or impedance but I was thinking about 3 meters of this: http://www.batt.co.uk/upload/files/audiocabletwopairtwinshielded8728_1642771387.pdf which has a the following characteristics
Capacitance (cond/cond): 115 pF/3
Capacitance (cond/scrn): 203 pF/m
Conductor resistance: 56 Ohm.km
Nominal velocity of propagation: 66%
Nominal impedance: 50.0 Ohms
 
Why shielded cable? Unless you live under a radio transmitter.

In my head (a) if it's reducing noise for headphones that's a plus, (b) the cable can sit near the rest of the electronic clutter and (c) if I'm going todo it then do it completely and extend the ground shield from the metal case to the headphone lead.

I can see your point - plenty of noise electrically and the cable connectors would be common noise on both thus cancelling.
 
In my head (a) if it's reducing noise for headphones that's a plus, (b) the cable can sit near the rest of the electronic clutter and (c) if I'm going todo it then do it completely and extend the ground shield from the metal case to the headphone lead.

I can see your point - plenty of noise electrically and the cable connectors would be common noise on both thus cancelling.
Headphone load is too small for shield to take effect, it will also add more capacitance which make the amp slightly more unstable.
 
Even unbalanced connection. Dynamic (4r-600r) headphones almost never pick-up stray electrical signal. I spent a career with headphones, often disconnected and laying on high power speaker lines or power lines. Try it.

You "can" get pickup. When I lived near a high power radio transmitter it did not take too many feet of extra cable and dirty contacts to hear beer commercials. But that was crystal earphones which are very different from modern headphones.

I'm not so concerned about capacitance. If a room-long cable's C kills a headphone amp, it wasn't any good anyway.