I measure approximately 15V as I should, but when they're of I measure -9V.
Do you actually mean -9V? How can you measure 15V when there is only 9V connected to it according to that diagram.
What you are saying makes no sense at all, therefore I suspect you are not measuring what you think you are.
Note when the transistor Q1 is off then effectively there is nothing to measure and so the inputs to the meter floats. You will not see minus nine volts but you might get a bit of pickup. This is true because the load is sevral LEDs so it is not a linear load.