Before going through all that, there are a couple of things you can check first.
Open the same dialog again, and consider all the other options in the dialog, below where you choose Length or Area. They configure the text which is created by the extension. (You don't have to keep the text. After you get the measurement, you can delete the text.)
For example, here's what happened when I tried to measure a simple gear type of shape (attached).
For best results measuring a closed path with a gear type of shape (since you used "perimeter" I'm assuming they are closed paths??) I would suggest using the Fixed Path tab, and the Center of Mass option there. Then below the tabs, you'll want to pay attention to the Offset value.
Theoretically, the text could be appearing outside your screen area, or depending on how it's configured, perhaps just can't be displayed. For example, using Text on Path, Start or End, if the offset value is negative, and smaller than the diameter of the object, it's geometrically impossible for it to be displayed (let's say, a 10 px diameter gear, with -20.0 offset - I'm not sure how that could work).
Hhmm, I guess I'm getting some unexpected results with the Offset value. For some reason, smaller values are putting the text further away.... I should probably also mention that these are relatively new options in that extension, all of which I haven't tested yet. But at the moment, I'm thinking Fixed Path > Center of Mass, should at least allow you to find the text.
Edit - Could there be a bug in this extension? Also, why does it have a separate Scale Factor? "convenience" as an answer, doesn't quite seem to fit, to me.
