The dryer is a 1962 800, top of the line as far as 1962 models go, but I think the 1961 LKs may still have been available for true TOL buyers on the sales floor. The 1962 800s were the basis for the very popular and long-produced 1963 Kenmore 800 Lady Kenmores. There were only a couple slight variations in features between the 1962 800 and 1963 800 LK washers, but the dryers I believe were identical in features, probably identical in everything but console cosmetic pieces. The 1962 800s are quite rare. I have only seen a couple dryers show up here over the years, and no washers. One of my childhood neighbors had a white washer (suds model) and it like all those in our area made it into the mid to late 1970s before worn bearings caused most of them to be replaced.
The washer is a mid-80s belt-drive. There seems to be a lot of these still hanging around. These were made during the conversion to the direct-drive models, which happened from 1982 (ish) through 1986. During that time, Kenmore's standard or reqular capacity models were mostly direct-drive, while all largest capacity machines remained belt-drive. In fall 1986 the direct drives finally took all the way over.
This washer is a 1983 model, but was produced in large numbers into Fall, 1986.
This is an interesting combination of machines. These dryers aren't known for their huge capacity, or their drying efficiency. I wonder how it handled full loads from that washer?