How can I safely extend my surge protector's cable?
If I understand correctly:

You have a surge protector that has an F type plug on it. This plug has a ground connector, but currently the ground is not in use, because you are plugging it into a K type receptacle (a receptacle that's compatible for the power pins but which doesn't have the right connections for ground).
You would like to extend the cord. As an added benefit, you would like to make the ground connection, and feel that replacing the cord with one that has a K type plug on it would solve both problems at once.

Depending on the construction of the surge protector, this is probably &quot;fine&quot;. Your challenges will be finding a cord of the appropriate length with the plug you want, and opening the surge protector to change the wire. If the surge protector uses screws for the case, opening it should be straightforward. If it's glued, as some cheaper-built ones would be, it's probably not worth the effort.
Inside the surge protector you may find that the cord is soldered, or it may be attached with screws. If the latter, replacement is very easy. If the cord is soldered, that could be harder; high-current wires are soldered with higher-temperature solder and with a larger amount of solder, which means a normal hobbyist soldering iron might not work that well. I've managed to deal with those kinds of connections, but it's a pain and is messy (I'm not that skilled at soldering, and it's easy to overheat neighboring components, circuit board, etc. in the process of trying to disconnect the old cable and solder back on the new one).
I wouldn't do any of that.
For your situation, I would instead just use an extension cord. You want something heavy-duty, with 12 gauge wire, not a light-weight lamp cord extension. With an appropriate-sized extension cord, there will be no problem at all.
I don't know what the types of extension cords available in Denmark are, but given the mix of outlets in the country, it's possible you can find an extension cord with an F receptacle end, and a K plug, solving both problems at once.
If not, then surely there are electrical adapters available which can convert the F plug to a K plug, which you can use in conjunction with the extension cord to provide the added length and the ground connection you want.
That will be just as safe, and doesn't involve having to modify the surge protector at all. It's not quite as pretty, but there are fewer opportunities to mess things up. To me, that makes it a much preferable solution. :)