Whirlpool electric dryer works on 120 but not 240
You can't use a 3-wire socket

You're not allowed to use a 3-pin socket anymore (been outlawed since the 1990s). You need to use a NEMA 14-30 cord/plug and socket.  There is a procedure for changing this on the dryer that involves removing a jumper between neutral and ground.  

If you insist on using a 3-wire connection to a dryer, you need to put it on a GFCI breaker.  Those cost about $80.  I recommend a 4-wire connection. 

Use of /2 cable (2 conductors + ground) is not allowed

You need /3 cable (3 conductors).  Dryers need neutral.  And you cannot use the bare wire in the cable as a neutral; that can only be used for safety ground.  That's because neutral handles current under normal conditions, and that bare wire is not safe to use that way. 

Once you have /3 + ground cable, how to hook up the NEMA 14-30 connector is obvious.  

Your cable must be 10 AWG not 12

If you are retasking a 120V circuit to be a dryer circuit, it is surely 14 AWG or 12 AWG.  Those wires are too small to support the heavy current draw of an electric dryer.  

Your breaker must be a 2-pole not a duplex

... and it will not fit in one slot (space). 

Fair chance you have a "duplex" or "double-stuff" breaker, with two handles. The breaker fits in one space. The dead giveaway is that the handles throw independently from each other, which means it is absolutely inappropriate for a dryer.    Dryers need a breaker with common throw on the handles, and also a feature called "common trip" which assures both sides trip at once. 

Further, dryers need a 240V breaker.  By definition, a 240V breaker must span across two "spaces", or it wouldn't have 240V!