|The piece that broke off may be inside the drive itself. Remove the drive module and take a look inside it, if you can open it up.|
I haven't taken a laptop CD drive apart. If there is something inside the drive that holds down the top of the CD when it is inserted like there is in a desktop CD drive or most external CD drives, anything that is very close to the diameter of the center hole in a CD will do, as long as it is accurately centered, because the piece in the center of the "spindle" as you call it that turns with the motor does not hold down the CD itself - it just centers the CD. If there is nothing like that in your laptop CD drive, then the piece in the center of the "spindle" must also hold down the CD.
Compaq and IBM do not make the CD drives - they are supplied to them OEM from CD drive manufacturers.
You could cut out the center CD retainer from a CD case and glue it to the center of the "spindle" - the bigger the retaining leaves the better, nylon would be better than the typical hard plastic but is pickier to glue on and get to stay on.
If the drives are made by the same manufacturer, yes, it's possible you could remove the "spindle" from one and install it in the other one but the odds are against that.
There are only so many CD drive module types for laptops - it's more likely the whole unit may be interchangable than the "spindle" of one drive would fit in another drive but the odds are against that too.
CD drives are expensive enough you may be able to get replacement parts. If you can determine which manufacturer made the drive - sometimes that info is in the parts lists for the laptop - you may be able to order the pieces(s) you need.
E.g. for Compaq and HP, the parts info is here:
Phone laptop repair places to see if they have suggestions or parts.
High speed CD drives do not last forever anyway - usually the first thing that kills them is the bearings in the motor wear to the point where there is too much friction in them and the motor can no longer spin as fast as it should, and eventually it can no longer spin at the minimum speed, or won't spin at all.
If you can't afford a new drive, you may be able to find a used one, though I don't recommend that, at a place like this - use the Compaq part number, or the manufacturer's model number:
If you can afford a new drive but don't like the higher HP/Compaq price, search the web using the part number of the drive or the manufacturer's model number - there are lots of less expensive drive modules available on the web that will fit the computer.