Stock wiring. Had them for a while and this issue with the blue light has just occurred. Will check the wiring but as all the lights work I assume that everything is fine and it is only the led bulb thats blown...or whatever leds do.
Hmm.. I bet you have another issue and this is just the first symptom.
The high beams on the ST1300 are funky as one is powered directly from the Hi/Lo switch and one comes through a relay that is driven by that switch. I don't remember what side drives the bled LED for the hi-beam indicator.
But... a bigger issue is that the ST stock wiring can not reliably handle the power of HID's over the long term. A few folks have done it, or tried, and they always end up having problems eventually. When HID's first fire they will pull up to 10 amps each.
To reliably drive them you need to put in a dedicated wired circuit for the ballast. A few of us add in a time-delay so the lights do not try and fire till the bike has been started. Once the alternator is running and can take the load is when they should be coming online. Better for the battery and better for the ballast this way.
You can use the stock Hi beam wiring to drive the hi/low switching of the HID's. (I'm assuming you have a bi-level HID kit.)
But as far as the indicator, check both fuses, check the relay and then if all that is good, measure if you have voltage at the hi terminal on the stock headlight plug. I'm betting one is out.
LED's can fail, but the one in this instance it's pretty rare.