Well the LM3915 data sheet says:-
Output current programmable from 1 mA to 30 mA
So your 60mA is too much. However, 60mA is not a high power LED and so is quite easily controlled using a resistor.
You can still use the LM3915 but replace the LEDs with resistors. Then connect the outputs to a 74LS04 inverting buffer ( 6 in one IC ). Then run the output through a resistor and on to the base of a transistor. The emitter to ground and the collector to the current limiting resistor. The other end of that to the cathode of the LED and the anode to the +ve supply.
The transistor only needs to be a small one and the TIP122 is way over the top for this. Any NPN transistor can cope with 100mA and you need only 60mA so get the cheapest one available at your local supplier.
Alternatively the TPIC6B595 power shift register can power these LEDs directly