Your trig is slightly off: you need to illuminate your box corner-to-corner, which is 7.5 * sqrt(2) on the diagonal.
The "angle" quoted for leds is usually the "half-brightness" angle, i.e. the intensity is half the maximum at those limits. That still won't make it evenly lit. The deeper the box the better and the wider the angle the better, plus your blob of hot glue, and paint the inside walls of the box white.
As for controlling your leds, there are about a squillion ways to do this.
TLC5940 not needed, as you don't need to fade your leds.
Question 1: to multiplex or not to multiplex?
No multiplexing: you need 8 x 74hc595 shift registers, daisy-chained, 3 Arduino outputs required. Very simple to understand, wire up and code, but alot of chips.
Question 2: DIY multiplexing or "does-it-all-for-you" chip?
"Does-it-all-for-you": max7219. There is an Arduino library for this, although you might not need it for a single chip. This chip does 1:8 multiplexing.
Question 3: for DIY multiplexing, what ratio? 1:8, 1:4 (even 1:2). It all depends how bright your leds are and how bright you want the table lit. The higher the multiplex ratio, the dimmer the output, for a given current. You can boost the current, but there will be limits imposed by whichever chips you choose.
Some combination of one or more 74hc595, possibly with ULN2803 or a bunch of transistors, or perhaps TPIC6B595 (which is like a combination of the two). Arduino pins: anything from 3 to 11 depending on the design.
Once you have chosen the leds, let us know their spec. (forward voltage & max current).
Don't let all these decisions put you off. Even 1:8 multiplex ratio will probably be bright enough. I built an led 4x4x4 cube with max7219 and it was plenty bright enough.