And now I know what's missing. I knew there had to be something to engage the wheel with the shaft...there is no "little clutch washer." Nothing between the hand wheel and the center knob and there is a little bit of play, back and forth, of the hand wheel between the machine and its knob.

Does it have a wooden cover that is curved across the top and smooth?

Yes, and the case is in excellent condition. I have another, a 99 with the same kind of case, but the veneer is splitting on the sides. I would really like to see this little Baby sewing. I was thinking, I could probably check for the washer on the 99 to see if there is one and see if it fits on the 128

yep this is a Singer 128-23. Made in the 50's. Look at the Singer badge on the front. If it has a blue ring around the outside edge, it is after 1951 centennial.
I have one just like this. Mine was worse for wear and I made mine into a hand crank. But it is a fun machine. Good Luck with it.

No blue ring around the outside edge. Its serial number is stamped unevenly and has proven difficult to read. I had it sewing today...stole a clutch washer from another machine, a Kenmore that's not currently working and that's what it needed. It's still a little stiff, but it's working. I got to sew on 3 different machines today! Fun!