I was not paying attention to my intrigue level and lost my capital city to a Coup. I had plenty of units near by so went in to get it back. The city went to 1hp, but that was it. I was unable to take it back. No amount of force would allow me to move in. What am I doing wrong?
Ranged units cannot capture cities themselves; they can only lower their HP. If you are using only Ranged units then it is impossible to capture a city.
The way you would need to capture the city is to attack with a Melee Unit. Melee Units can capture a city when they attack the city and the city has a low enough HP. If a city only has 1HP, then even a basic Soldier should be able to capture it.