Best (or most affordable way) to get from Gare du Nord station to Port-Royal Hotel?

You can take as much stuff as you want.

Assuming that you don't want to do excessive walking, you need to get to the Les Gobelins metro station. From Nord, it's Metro 4 to Chatelet then Metro 7 the rest of the way. The Chatelet change isn't a barrel of fun.

If you could walk about a half mile with your junk, Metro 5 out of Nord would dump you at Place de Italie with no changes and you could hump the rest of the way.

Either way, the metro ticket is less than two euro (the same ticket will do either).

Or you could walk over to Gare de L'Est from Gare du Nord and take the Metro Line 7 to Les Gobelins. It's about an 8-10 minute walk.

Ed, at least that walk from the Place d'Italie station is down hill all the way. But, even empty-handed, it's still a minimum of 15 minutes walking.

Actually, I'd bet it's a bit less than half a mile. I was wandering those streets when I was still in short britches.
My sustained flat hump speed is three point five per hour with anything less than forty pounds if its on my back.
Call it less than nine minutes.
Gross prior imprecision regretted.

I hope you will pack light, for you own benefit in a variety of "budget travel" situations. I've taken my one carry-on sized backpack and small, cross-body day bag onto innumerable public buses and subways, including London and Paris. It's less fun with rush hour crowds.