I have recently entered the 3D  printer game with a old Geeetech A10 Pro purchased at Goodwill.  There's a lot to learn!  One of things I've learned is that it's sometimes necessary to clean the hot end when something goes wrong.  One of my first troubleshooting finds was that the hot end fan shroud had been melted during its past life, and had a tendency to sag the print and drag it around the build plate.  Which of course left the hot end coated in PLA.  A bit of Dremel tool action solved the problem.

Seeing the hot end involves a lot of neck craning, trying to see the problem and clean up the mess.  Of course, the solution to a 3D printer problem involves a 3D printed solution!

I came across some round 50mm diameter mirrors at a craft store (labeled as 2" – lesson learned: check dimensioons!) and the idea for an inspection mirror was hatched.  I used OpenSCAD to design an inspection mirror that sits on the build plate under the hot end and provides a clear view from a comfortable working position.  Excellent!  My OpenSCAD file and STL are below.

 But this still left one problem.  It's hard to get much light on the hot end.  Lighting it up with my phone either directly or by directing the light into the mirror worked OK, but wasn't ideal.  The inspection mirror needs an LED to light up the work area!  A T1¾ (5mm) non-diffused white LED provides an intense narrow cone of light.  That would be perfect.

I had to break out my geometry skills.  The center of the mirror should be directly below the nozzle, and this should be the focal point  of the LED.  But the LED is offset from the center of the mirror, so it needs to be angled less than the mirror angle..  The illustration shows my graphical calculations.  A 10° offset will be needed.