Hi,
Had the same problem with my AMILO XI 3650. Seems to be a very common issue as i found in the net.
Its the the receptacle for the charger plug that gives a bad or no contact. Since i could not open the laptop (some hidden screws ??), i just removed only the cover that gives access to the fan and DRAM slots.
If you remove the fan (very easy - only 2 screws) you can see this receptacle behind.
The problem is the center connector (i.e. + pole) which is neither welded or soldered to its base platen but only pressed in. So, after - in my case - some years of use this connection gets loose due to mechanical stress ( you pull on the cable of your charger now and then) and the consequence is a bad contact.
The plastic of my charger plug literally melted because of this
bad contact with its high ohmic resistance. After a while, my laptop was completely dead - as yours.
Since you cant reach this receptacle with a solder iron by just removing the fan, i drilled a hole of about 10mm in diameter just above the receptacle and soldered this rotten connection.
Everthing is working fine since more than 6 months now !!
P.S.:
Its also a good idea to replace the plug of the charger and clean the contacts of the receptacle carefully - eventually with some sandpaper.
Hope i could help you - good luck.