I decided to try making more embroidery/string art. I didn't follow any specific pattern, I just used trial and error since it's very easy to fix your mistakes.
I used a 5/32 drill bit to make my holes, which was a lot larger than my first attempt. The larger holes are much easier to work with. I could go through the same holes multiple times and I didn't need an upholstery needle. Instead I used some Mod Podge glue on the tip of the string to make it stiff and just strung it though the holes.
I included a picture of the back (below) so you can get and idea of what it looks like. I tied some string to each other to hold them in place. I also made sure to double up the string in the back only, so that the front looked neat.