Sounds wierd... If it's still running when you turn it off, have you got a short around the ignition switch? Something that's allowing current to flow across the switch even when off?
Unless the engine was running, you woudn't notice it untill you had a flat battery.
Has the old instrument cluster got some sort of relay in it that you're missing when you connect up you new dials?
Not the same as your issue, but I had a problem with my Sportster as I'd snipped a wire in the harness that I didn't think I needed, which was fed from the old BMW instrument cluster - until I realised it was the main relay energizer! It would never have started without it...
I'm sure some of the more knowledgable auto electricians will be along in a moment. I'm still messing with my knitting as they call it.