This picture shows an adapter board which is intended to go into a solderless breadboard. The right-angled header pins at the bottom must be of different heights in order to enter the PCB at different places but end at the same place on the bottom of the object. How do I find right angled headers of two different lengths, like this? Thanks.

 

How do I find right angled headers of two different lengths, like this? Thanks.

Have you tried the parametric search engine of your favorite distributor? Digikey and Mouser let you filter by length.

 

These also come in a two row version for this application.

 

Here is what I think you want: http://suddendocs.samtec.com/catalog_english/mtsw.pdf with options -Q (double row, 0.2" spacing) and -RA (right angle). There are other options you need to decide on. Document found here: https://www.digikey.com/products/en...612&quantity=&ColumnSort=0&page=1&pageSize=25

You may also opt for a 3-row version and remove the middle row.

 

How do I find right angled headers of two different lengths, like this?

If this was for one or a few boards, I'd just buy a standard double row right angle header and modify them. It can be a pain, but I've done this. Not because I needed something like your application, but because I had 2x17 and I needed something like 1x5. I used cutters like the one shown below to customize length:

Then I pulled out half of the pins, cut the polyester bit, and reinserted the pins.

If you're okay with the lengths being off by 0.01", you could order standard single row parts: