I found another thread where someone tried to install sqlite3 manually. I tried those instructions and got the following output:
ruby 2.0.0p648 (2015-12-16) [i386-mingw32]
C:\Windows\System32>c:\ruby200\bin\gem.bat install sqlite3 --source http://rubygems.org --no-ri --no-rdoc
Fetching: sqlite3-1.4.2.gem (100%)
ERROR: Error installing sqlite3:
The 'sqlite3' native gem requires installed build tools.
Please update your PATH to include build tools or download the DevKit
from 'http://rubyinstaller.org/downloads' and follow the instructions
at 'http://github.com/oneclick/rubyinstaller/wiki/Development-Kit'
I tried following those instructions to find the right devkit but none seem to match this version.
Finally got mine working. Turned off UAC, deleted everything I had, and installed everything to the C: drive, even though I normally keep NOTHING on C besides Windows. Fixed the problem entirely.
I left everything in default directories, which were all C: but still nothing. Did you have any files from previous attempts sitting around somewhere that you missed previously? And you're just using the launcher that installs Ruby 2.5.5 right?