Fault finding is a process of eliminating / ruling out one thing at a time.
On your K5 Wee, you can turn the hot idle speed up by adjusting the remote idle screw in the Y of the frame by the riders left knee. See the K4 manuals available under
https://www.v-strom.co.uk/downloads.htmOnce the idle speed is correct, I suggest that you test the charging system.
A few tests to rule out the battery as a first step:
1) battery terminals clean and tight?
2) perform a load test on the battery - the load test method of taking a voltage reading of a rested battery as a before reading and again while the motor is cranking, will help to determine whether the battery needs replacing. Another alternative is to simply have a battery center perform a load test and give their verdict on the battery condition.
Next step is testing the charging system:
1) take a voltage reading at idle speed and then another reading at above 4,000 rpm. Share these readings, so that the members here can offer advice or comment on the readings and the next steps to perform.
2) the next steps would be testing of the stator output voltage, stator coil resistance testing, stator earth leakage testing. The stator stesting is discussed in the service manual.
3) the final step in the charging system would be the voltage regulator testing.
The flow chart (fault-finding-diagram charging system.pdf) that is attached in the opening post of
https://www.southbayriders.com/forums/threads/145945/ may be of interest.