did you rinse it thoroughly under running water BEFORE you applied the lighter fluid? If not, your problem may be soap residue on the movement, which in turn could clog up the balance wheel causing it NOT to function.
I would suggest re-cleaning it with the dishwashing liquid. THEN, thoroughly rinse it with clear water from your tap, holding the movement under the stream for a minute or two to be sure that ALL soap residue it rinsed away.
Then, DRY the movement thoroughly. If you have a can of "canned air", the kind used to clean computer keyboards and electronic gadgets, and blow out the water. You might even use a hair dryer on low heat to be sure the movement is dry.
Then, you could rinse with lighter fluid, which would assure all water is gone. After that, oil the key pivots, etc, as suggested by RonD.
Hope this helps.