Added LEDs to the ciruit and the code to help with debugging. This has confirmed that we are entering the while loop and that we are going through the charging cycle. The oscilloscope is showing weird behavior that could account for the fact that we are never entering the timer1 interrupt. We aren't fully discharging and as such we are never generating a rising edge on the comparator in order to get the timer 1 capture.
I changed the timer1 ISR to start discharing after the match was made. This improved the oscilloscope waveform to something more desirable than before. However, the LCD is still never changing, which makes me think that we have an error in our calculation step. Ask Prof. Land what he thinks of the oscope waveform to officially rule out our timing as a source of error, but I think it's pretty good as is.