The part you see is called the planetary, it has two shafts, the long one is called the center shaft and if it is falling off you need a new planetary, the second shaft (shorter) is called the agitator shaft (the beater attaches to it, if this is the culprit then1 you are not attaching the beater correctly2 the locking pin that the beater attaches to to hold it in place is missing3 you have way too much "gunk" build up on the agitator shaft and the beater is not attaching correctly4 if the beater falls into the bowl with the agitator shaft still attached you have three additional problems
a) the snap ring a top the agitator shaft has broken off or come off which in turn means that
b) the gear at the top of the shaft has come off and may very well been knocked around and possibly damaged beyond future use which means that you will need a new one and also
c) the shaft pin that holds the agitator gear in place has come out or been broken alsothe fix is easy and your parts cost $ will be from $30.00 to $80.00 dollars depending on which parts you need. Now before you get palpitations, cost are listed as a possible need1 new planetary $30.00, you may not need2 new agitator shaft and gear assembly $27.00, pretty sure that you need but maybe not3 new gasket $6.004 new gear grease $6.00 KA says 2 oz. (don't buy a 2lb (26.00 bucks) can when you only need 2 oz).Without more clarity or a picture I can't help further. but before you start tearing you machine apart go to the KA website and get a parts list for your mixer http://shared.whirlpoolcorp.com/product_literature/search.jsp?siteCd=KAD
let us know how you fare on this or better yet get us a little more info and we can help more.