What is that supply capable of 29 mA? That can't be used for anything.
External 5 volt can't be connected to Vin, only to the 5 volt pin. Remember to disconnect it when using USB connection.
It looks like a wiring issue - those LEDs apparently draw 18mA each, so I would expect at least a few more than seven to be manageable. You might try only lighting one LED on each jewel to be sure.
How many LEDs do You want to turn as a maximum?
A few, say 5, maybe 10, will be possible from the UNO 5 volt.
Can You supply a link to the datasheet of those flowers?
Shifting 14 bits ought to cover all of them.
A tiny schematics made by pen and paper might be useful.