The LCD and the e-ink keyboard both use 60 pin 0.4mm pitch connectors. These are the same flavor of connector as the Intel Edison uses, it just so happens! The Edison uses the 70 pin version, which is fine. I'll regardless need to have various resistors/capacitors, I'm sure.

For reference, they are Hirose DF40 connectors: http://www.mouser.com/ds/2/185/ed_DF40_20140305-337786.pdf

Mouse has e.g. this pair of 3mm tall connectors that are readily available:
http://www.mouser.com/ProductDetail/Hirose-Connector/DF40HC35-60DS-04V51/?qs=sGAEpiMZZMvffgRu4KC1Ry7cu3RxfNlTTj5mBFiCV9w%3d
http://www.mouser.com/ProductDetail/Hirose-Connector/DF40C-60DP-04V51/?qs=sGAEpiMZZMvffgRu4KC1Rw0uA%252bB%2fcqq%2f9Sn%2f4WIT5YQ%3d

There's just enough labeling that I know which pins are which. (Only one is labeled via screen printing, but both have the same label on the header, which tells me the connector orientation.) The mainboard side has the "header" half of the connectors, while the two ribbon cables terminate in the "receptacle" half of the connector.

At a later point I will get better pictures of the PCB and attempt to identify the components near the connector. I can already see where the power lines probably are; thicker traces and the telltale color of SMD capacitors.