DIY Inductor: Solve for number of turns without knowing the length?
You have experimentally derived the equivalence that 113 turns yields one inch of coil length, yielding the conversion factor (1 inch / 113 turns). Obviously this tacitly depends on the coil diameter, but is a sound figure for the diameter under which you established the measure.

If you have 113 turns that is one inch. If you have 226 turns that is two inches. and so forth. So Length definitely equals number of turns / 113. Looks to me like you can safely substitute N/113 for l. Just keep in mind this doesn't generalize because of the implicit dependence on diameter built into the 113.

You should probably just ask the second part of your question as a new question. It's way more interesting than the part I'm answering here :-).