1) Your circuit only needs a few tweaks. Use ONE nonpolar capacitor, with both LED strings in parallel in opposite directions as you show them.
2) Calculations for the capacitor value are complex (pun intended); your value looks like a good start but fine tune it after measuring the actual current. Beware that cheap NP electrolytics have wide tolerances so plan to experiment.
3) All of the LEDs in each string will get the same current. If the string gets 20 mA, then the white LED just won't be as bright as you expect. Hopefully no big deal.
4) As ag said, the LEDs in opposite directions should protect one another.
Your transformer is on the edge, 200 mA at 12V is 2.4W. If the motor really does draw 2W, adding 20 mA (0.24W) load brings it close to its full rating. This isn't usually good. Watch for signs of overheating, or as ag said, get a bigger one!
Did you know your LEDs will only be operating at about 20% duty cycle? Each string only works on its positive half cycle, and even then only when the sine wave is enough to turn it on. They won't be very bright at all.