I go through inkjets like candy. I'm currently on an HP Photosmart 7850. I also have a wide-format HP Deskjet 9800. I can't remember if I did the Narnia warrants on the 7850 or the printer prior to. I usually get a new printer after two sets of ink.
The Narnia warrants were on special paper (as is the Hogwarts invite). The trick is to massage your settings -- ink amounts, overspray, etc. I also play with layers and such in Photoshop, deepening and lightening pen strokes, sometimes adding under layers that might add more randomness to the pen.
The hardest part of this replica is actually the envelope -- it's very thick and tends to jam the printer. Luckily, the die I had made had score lines for the envelope folds, so that takes down a bit of the thickness.