You can't draw any, it's an open collector output. It only sinks current.
Each "driver" inside the chip can sink a maximum of 500mA. The datasheet doesn't specify a maximum current thru the ground pin that I could see.
the device will get hot in a hurry if you try to sink 500mA on all 8 drivers simultaneously.
TLC5940 sounds good, but I'm worried about the drive capability of only 160 mA. If I put four or five 20 mA LEDs on each output I'm again approaching total current that may be too high.
It looks like the (ULN2803A) package can dissipate a maximum of about 1.8 watts total at 25C ambient (room temperature). You might be able to increase that a bit by using heavy traces on the PCB, or as heavy copper wire as you can if you are hand wiring. But if you push it to it's maximum, it will be too hot to touch.Just a suggestion here, but I would keep the total package dissipation to below 1 watt. The spec sheet gives the Vce as around a volt if you keep the current on any given pin below 300ma and that would suggest a maximum, if load is spread out across the device, of about 1 amp total. If you limit your LED drive current to 10ma, that would suggest you could run 100 LEDs per device with no more than about 30 on a single output. For 20ma, cut those numbers in half.As has been said , if you needed more LEDs than that you could stack them in series and use a higher drive voltage. As an example, if you re using white LEDs with a forward voltage of 3.5V and drive them with 12V you could use 3 in series with a 150 0hm resistor to drive them all at 10ma. But since you said you won't need more than 15 per output, that shouldn't be a concern.
I think it's safe to consider the 1.8W and 25C to be mutually exclusive unless liquid cooling is involved.
I would also add that most modern LEDs shine very brightly on just less than 5mA. They last forever that way too. 25mA just leads to short lives.