After attempting the reset a second time and it not working, I continued my search on Google for the answer to the problem I was having and was able to find the solution. On a couple of websites it had been noted that uncleared paper jams would cause the toner light to blink. There was no paper in the printer's various jam locations, so I carefully examined the printer with a high powered LED light and found a barely visible scrap of paper sitting behind the rollers that feed printing area in the main body of the printer. I was able to just fit a pair of tweesers between the plastic and the rollers to remove the bit of paper and found it was sitting directly on top of one of the paper sensors. After removing said piece of paper I powered on the printer and it is now functioning normally.
So for the future reference of whom it may concern: if the toner light is blinking on your HP LaserJet P1606dn and your cartridge is not being recognized, double check that no bits of paper are somehow stuck/jammed inside of your printer. A very minor paper jam may be your issue. (And be careful sticking anything inside your printer! Don't accidentally damage your printer!)
And thank you TECH58 for your time and quick reply. It may not have been the solution but it is nice to get something back when you are not expecting it.