Using LEDs in an output of a power supply with a voltage divider
It's not much of a power supply- you won't be able to draw much power from the middle voltages without dropping the voltage significantly. It would be better to switch resistors on something like an LM317 (and make special accommodation for the 0V by shorting it directly to ground since you can't get below 1.25V with an LM317- a typical situation with regulators in general, though the reference voltage may vary). 

You have an interesting problem in that you have expanded the scope of the project (adding LEDs) in a way that will be more difficult to accommodate than if you had considered it to begin with. It's not an unusual situation to occur in the real world. 

One rather crude method would be to use the switches to operate telecom style DPDT relays and use one contact for the voltage selection (by whatever means) and use the other to activate the appropriate LED. You would need 6 relays and one resistor. 

A better way might be to use a couple analog switch arrays such as CD4066 quad switches, and control both the LED and the analog switch with the mechanical switch. The analog switch would select the desired voltage.