How can I connect a sub-panel to an unused 60A breaker in another panel?
Make this a 120V only subpanel

Since the original circuit was 240V-only (hot, hot, ground), you're short the neutral wire needed for having both 120 and 240V at the old furnace's location.  Since you only need 120V here, though, what you can do is install a 120V-only subpanel; however, that limits you to 50A as 120V-only (by design) subpanels don't come in larger sizes, and trying to use a larger subpanel gets confusing, what with the potential for skipped slots and all.

You'll need a GE THQL1150 breaker (to replace the THQL2160 that's in there right now), a TQLFP1 filler plate to fill the hole left by the other pole, a Square-D QO24L50TTS panel for your subpanel, a suitable NM clamp for the cable coming in, and appropriate QO single pole or QOT double-stuff branch breakers in 15 and 20A, as well as a torque screwdriver or torque wrench that goes up to 50 inch pounds or so as all lugs must be torqued to their manufacturer specification (110.14(D) in the 2017 NEC).  Replace the THQL2160 with the THQL1150 (with the main off) -- the black wire that landed on the THQL2160 lands on the THQL1150's lug, the white wire that landed on the THQL2160 gets moved over to land on the neutral bar (torque it to spec too!) as it will be a neutral now, not the other hot leg, and the bare wire stays put as it will continue to be a ground.  Snap the TQLFP1 into the hole in the deadfront left where the other half of the THQL2160 was.

Once this is done, leave the new branch breaker off, button the main panel up, and turn the main breaker back on.  You can then mount and install the QO24L50TTS in an appropriate spot; once it's mounted and the cable fed in through a KO and secured, you then can terminate the black wire from the cable to the main lug, the white wire from the cable to a lug on the neutral bar on the far right, and the bare wire from the cable to a lug on the ground bar on the far left.  Make sure to torque the connections to spec; there will be no bonding strap or bonding screw to pull, as the QO24L50TTS is not suitable for use as service equipment to begin with.

Then, you can install the appropriate OQ or QOT single pole breakers (QO115/120 or QOT1515/1520/2020 depending on the mix of 15 and 20A circuits you need, you can get 4 circuits max using double-stuffs in this panel) and wire up your 15 and 20A circuits to the subpanel.  Note that if you want ground-fault or arc-fault protection here, it will probably be easiest to use a short conduit nipple over to a box with a deadfront GFCI, AFCI, or DFCI in it instead of crimping your circuit count with a GFCI, AFCI, or DFCI breaker.