Electronic – How to determine what capacitance is required for eliminating humming noise

The culprit is not the LEDs themselves, they're harmless, but the TLC5940s, which switch at high frequency to control LED brightness through PWM. You can't filter the PWM outputs (you can, but then the brightness control doesn't work properly anymore), so that's out, but you can try to do something about decoupling power supplies. Not guaranteed to work, the fact that the scope's probe picks up the signal unconnected indicates that it's probably radiated, but it's worth trying.
Decouple the TLC5940s properly. They have to provide a lot of power, so that means 100\$\mu\$F, 1\$\mu\$F and 100nF all parallel on the power supply for each device, the smallest value closest to the pins.
Decouple your 5V power to the microcontroller properly: 100nF close to the pins.

A transformer, as you well know, is made up of two or more coils around a core of ferrous material. That ferrous material is not a solid lump of metal, but a series of plates laminated together with adhesive. This is done because:

Early transformer developers soon realized that cores constructed from solid iron resulted in prohibitive eddy current losses, and their designs mitigated this effect with cores consisting of bundles of insulated iron wires. Later designs constructed the core by stacking layers of thin steel laminations, a principle that has remained in use.

-- Wikipedia

So you have lots of steel plates stuck together, but not only that:

Each lamination is insulated from its neighbors by a thin non-conducting layer of insulation.

Lots of metal plates, each with an induced magnetic field. That magnetic field acts between the adjacent plates stretching and squeezing the adhesive and insulation between them. Over time that adhesive starts to break apart and the laminated layers separate from each other slightly. This is the humming noise you can hear. It's always present, but once the adhesive starts to break it gets louder. These micro-fractures in the adhesive may not be visible to the naked eye, but in extreme situations they may be so bad the layers of lamination become loose and the transformer literally rattles as you shake it.

Also, the more current you draw through a transformer the larger the induced magnetic fields, and thus the louder the transformer hums (and the shorter its life span).