Reviewing this lesson the other day, I noticed something odd in the text. Why does it say "D'ardaigh sé an chathaoir le mé a dhéanamh níos compóirtí." rather than "D'ardaigh sé an chathaoir liom a dhéanamh níos compóirtí." ?

At a guess, I'd say that it's because "mé" is the direct object of "a dhéanamh".

In other words, it's (le) (mé a dhéanamh), not (liom) (a dhéanamh).

Makes sense to me, anyway....

--Al Evans

Exactly right.

[le] + [mé] = liom

[le] + [mé a dhéanamh níos compóirtí] = le mé a dhéanamh níos compóirtí

Similarly,

I made an attempt to do it.
Rinne mé iarracht ar é a dhéanamh.