Graphic designers have the same problem
There are a number of spider "doobies" on the market that sample colour on screen agains "real life"
trouble is ... the cheapest costs �170.00
you could search the websites for your monitor - look for colour profile *.icm files
or use a lot of ink by printing test strips ... then adjusting your monitor .... 'till you get a match.
then you change the ink ... and start again.
basically it's not going to happen
Design/printing costs �1,000,000s and uses carefully manufactured PANTONE controled inks
your HP uses 4 coulours ... and as you say - if you hadn't seen the original - does a reasonable job.
best you can do is compensate .. if your printer "washes" colours ... make them a little stronger on screen