Thermocouple Cable

You can't just calculate a number & apply it as a correction factor - it depends on the temperatures of all the junctions between dissimilar metals.

No. It is called a 'K-type compensating cable' for a reason. Were it to work with S-type thermocouples, why then would they call it a 'K-type compensating cable.' The name is a clue as to its intended purpose.

But! If you decide to use it anyway, you can save yourself some money and use doorbell wire instead. It's much cheaper whilst also giving you incorrect readings.

Type R or S thermocouple extension wire consists of a copper wire and a copper alloy wire since it would be prohibitively expensive to use platinum for extension wires.

The extension wire is actually compensating cable, designed to have the same EMF gradient characteristic at (relatively) low temperatures as either an R or S thermocouple.

The positive leg is pure copper and the negative leg is alloy #11, primarily copper, 97% copper with a balance of 3% Ni/Mn. It is a relatively inexpensive wire and can be used in environments up to approximately 204ºC.

In the absence of the correct Type S compensating wire, there is no means of determining the error created by the junction of K wire and the S thermocouple, so there's nothing to 'look-up' and correct for.