Hi will73,
I understand you have a Photosmart 5510 that is not printing black ink. I will certainly do my best to assist you with this.
It's possible that the non-HP ink has clogged the printhead of your unit, but there are still a few things you can follow to help solve your issue. First, I'd recommend you have a look through this Printer Does Not Print Black Ink or Color Ink, or Prints Blank Pages document. Second, I'll ask that you also have a look through this Fixing Ink Streaks, Faded Prints, and Other Common Print Quality Problems document. Between these two, you should be able to get the black ink printing again.
Hope this helps, have a great day!!