This is all abstract as I have no practical experience making lights, but:

True omni-directional light seems difficult. No single LED provides a beam wider than 70 degrees or so, so it would take some number of them arranged regularly (i.e. with 3D symmetry) to shine light in all directions, and it would probably require something like a frosted glass globe to diffuse their light evenly as each LED would not have even lighting across its beamwidth.

Geometrically, there are 5 kinds or regular 3D shapes (polyhedra) as noted

here. They have 4, 6, 8, 12, and 20 sides, and that's it.

You could use one of those polyhedra as a guide for pointing your LEDs inside the diffusing globe. Using such an arrangement with more LEDs would of course provide more even light sources for your diffusing globe. I would use low dome Luxeon type LEDs with no optics to get the widest beam possible.

As for 30 cd of brightness - the equivalent cd for a diffuse source is harder to calculate than for a point source, but you could estimate what you needed based on the lumens required. A 30 cd omnidirectional source is the equivalent of 30 theoretical "candles" burning. Each theoretical candle puts out 4 x pi or about 14 lumens. Therefore, a 30 cd omni source would require you to generate about 420 lumens with the total number of Leds you use. I'd make that 600 lumens to account for light lost going through the diffusing globe.

So you need to drive the total number of LEDs you used in your LED arrangement to generate about 600 lumens of light.

If you use Lux IIIs (no optics) driven at 50 lumens each (which is close to 1 amp, I think), then 12 of them arranged symmetrically (technically, in a dodecahedron arrangement) would put you in the ball park.

Of course, this is just theoretic. 12 Lux III's driven at 1 amp each will require lots of cooling which is not suitable for air cooling in a more or less closed sphere, so you'll need to do something about that. I don't know, liquid cooling?

All that said - and I con't know what your constraints are, but - I think a regular 60 watt bulb will put out about 850 lumens, more or less omnidirectionally except for the shadow of the base/socket (says

here). So could a 60 watt soft white bulb in a frosted glass globe also do? If so, it would be a lot easier...