You've already stated the most obvious "DIY" solution and the only one with a high percentage chance of actually working:

I'll rather use my outcamera flash to put on.

Unless you're skillful enough to almost totally disassemble and reassemble the camera and do some de-soldering and soldering to replace whatever part of the flash circuit isn't working (maybe it's a capacitor soldered to one of the camera's PC boards, otherwise the board will need replacing) it's the only viable solution. Since your external flash seems to be triggering fine, the issue is almost certainly in the internal flash's charging circuit.

Though 220€ seems a lot to spend to diagnose a D5500 only worth 210-230€, the service center has to still pay their technician for several hours of skilled labor to investigate the issue. In other words it will cost them more to diagnose and fix it than the camera is even worth.

So at this point if the external flash is not an acceptable solution for you then you're better off looking for another used camera that works properly instead of spending more to have this one repaired.