Hi Ken,
Flourescent lights will probably give you better results than the windowsill. That is how we grow all our seedlings and feel window light is inferior. Not only do you have cloudy day but many glasses are now coated with UV filtering coatings.We use fresh (they wear down after many hours of use even though our eyes can't detect it) flourescent bulbs and keep they very close to the plants. One warm and one cool bulb in the 4 foot double fixture is best. We just get the less expensive bulbs but still have great results. Replace the bulbs every year for serious growing conditions.
Set a timer for 16 hours a day to light your plants, plants need a resting time. One year we didn't do the timer and the plants protested. Honestly! They like to sleep just like you do.
Bye, Diane