We are talking about http://www.ti.com/lit/ds/symlink/tlc5917.pdf
It says on the front page: The constant sink current for all channels is set through a single external resistor.
So I flip forward to a pinout spec - page 4 - where it says R-EXT External Resistor - Connect an external resistor to ground to set the current for all outputs
. The diagram shows that as pin15. Zipping forward looking for a curve or formula for R-ext (it is a PDF and there is a search function in the Reader) - where I find on page 15 a nice chart and a summary: the default current is approximately 52 mA at 360 Ω and 26 mA at 720 Ω.
At least unless you program the VG (Havn't bothered to find out how)
SO if you put a 1K resistor it is down to 20mA.
Yes, reading spec sheets is difficult, one needs to try it, often, re-read it several times.