And now to the problem. When I hold down the "magic" button for more than 3 seconds, nothing happens. Not a single flash. When I press all the other buttons then the "TV" LED flashes.

The remote control is barely used, so it has not received any beer or soft drinks. The condition is like new. No battery acid from the batteries.

Is there anyone here who can have a solution to the problem?

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My Inconsistent fix ( NOT working anymore ; read next post):

I found a inconsistent fix / workaround:...
...you must try maybe lots of times... You should be a bit resistance of frustration...

You get no reaction at all, on pressing "Magic": Restart the remote.
Sometimes pressing "Magic" does not give at least a single LED-Flash.
A fully restart of the remote, may help:
1.) Put the Battery(s) out
2.) Press any normal key (s). (The internal capacitors are now discharged)
3.) Then, put the Batteries again in. The Led will flashes 2x times to confirm start of the remote.
?? Not sure: Maybe also shortly pressing any Key or the Magic Key seems to help ??

You only get a short led-flash, on pressing "Magic":
You must press very often, before pressing once long time (3Seconds). This maybe helps.
Long description:
Requirement: You remote must react (=flash LED) on pressing shortly on "Magic". Otherwise, this not helping.
You should do: Pressing at least one time shortly Magic. Then long Magic for 3Seconds.
Not working? OK. Try again and again. Try it 5x times or 10x times. It will working...
Example: Also try pressing at 2-5x time (or more) shortly Magic and then long Magic for 3Seconds.
There is no direct code or order or timing to do this (at least i don't know). Just try.

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Workaround:
This is a JP1.x remote. This means: You can program this remote via a computer. But i takes a bit of time for reading & learning. You also need little, very cheep hardware USB-Adapter. [Link: hifi-remote.com]

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Final repair :
Now I've found a way to repair.
You need a soldering station.
Replace a defect ceramic capacitor, with a new one.
I don't know to origin values of this part.
I just tried: 100nF (=0,1μF) with 25V is working. I think other values, should also work: e.g. 10nF or 1nF

Position of this part:
You found this defect ceramic capacitor directly above the "U3" label ( printed label on PCB).
This defect ceramic capacitor is parallel connected to the electrolytic capacitor "C5" (backside).
(( Unimportant background: It's seems connected at output-side of the IC ((a DC-DC converter?)). ))

The two electrolytic capacitors:
My first attempt to repair: replacing the two electrolytic capacitor on the backside. But, this doesn't helped. ... Anyway... Just in case... You can also replace these : "C5": 100μF 25V and "C1": 100μF 16V

(( common electric info: ( This may help you to find the replace parts. )
In this case: You can also use capacitor with higher values (nF/μF) ;; 
 An general: It's general/mostly allowed to use a capacitor with higher Volts(V), labeled on. ;;; ))
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