Instead of the SSH6868, you will need a shielded cable assembly to connect your 60xxE (E-series card) to the SCXI chassis. You will require the SCXI-1349 cable assembly. It has two parts 1)SH6868 cable and 2) an adapter to connect to the SCXI backplane.
As for the choice of the DAQ card,you may find the low cost approach severly limits the system bandwidth and overall accuracy. The low cost E-series boards only have 512 samples of onboard memory, as opposed to 8192 for the regular E-series. This makes controlling buffered analog input more difficult. In addition, you need to understand that the rated sampling rate for the board is based upon a single data channel. As you have 16 channels, your "best" sampling rate would be approximately 12.5 KHz per channel. The actual rate will be less. You have made no mention of the type of data you want to acquire (steady-state, dynamic, time domain, frequency domain), so, I can not give any better insight into the "best" sampling rate you could achieve.
Now, considering that you are mixing thermocouple (TC) readings with other inputs on the same DAQ call, you may find the 12-bit selection aggravates your data quality. When you acquire data using a multiplexer, the quoted accuracy for the device assumes that all of the voltage levels on the various channels are very near the same value. This allows a relatively uniform settling time for the input signal. It is quite possible that the TC inputs will be on the order of 2-10 mV while the other signals could range as high as 0.5 volts. Such a difference in input levels can lead to settling errors that are not obvious to the user, but can cause gross errors in the measured data levels. If you set the input range (only 4 for the low cost e-series) to +0.5/-0.5 volts, your resolution for the TCs will be about +/- 0.24 mV or about 6-10 degrees of error. (ouch!!). So you will be required to set different input ranges for the different input channels (0.05/-0.05 for the TC and +5/-5 or +.5/-.5 for the remaining channels. These multiple range calls can further limit your bandwidth by slowing down the "round-robin" interval between the data range changes. I know that your goal is low cost; however, you may find that the data quality is unacceptable. You will want to consider, at the very least, upgrading to the 6034E board. It is a 16-bit device (still only four input ranges); however, if you configure the board for only 1 input range, +0.5/-0.5 volts, your bandwith will improve and the TC resolution will improve by a factor of 16 times to approximately 0.015 mV (under 1 degree).