TLV5606 update frequency
The maximum serial clock (SCLK) frequency is 20 MHz and therefore the maximum serial bit rate is the same. This means that you can dump a 16 bit word at a rate of 1.25 MHz: -



Above on page 12 of data sheet.

Given that the settling time is not going to be so large that there would be no point shipping data at anything like 20 Mbps I would estimate that operating at 2000 samples per second is fairly easily achieved.

Page 1 says the settling time is 3 us in fast mode and 9 us in slow mode. So even in slow mode and adding an extra microsecond for delivering data, a sample speed of 100 kHz should be typically achievable.