DIL / dual-in-line header.
These are available with insulation displacement connectors (IDC) allowing termination to ribbon cables, and can be bought preterminated. Molex make these and also probably have solderable versions (you'll be sorry).
IF the pitch is 2.54mm, which seems likely, then an old floppy disk drive cable (34 pins) is probably an easy solution.
You MAY have a 2mm pitch header but 2.54 mm or 2.5mm is much more likely.
Spacing is usually either 2.54mm = 0.1" or 2.5mm = not quite 0.1".
The two are interchangeable (for non-purists) after a fashion for small numbers of pins but get out of alignment by 0.04mm per pin.
When measuring something like this it is a very very (very) good idea to give people as accurate a pitch measurement as you reasonably can. In this case you have 16 pins so there are 15 x pitch distances between 1st and last pin. Measure this distance and divide by 15. Or measure 10 pitches between pins 1 & 11 and divide by 10.
Even for 10 pins 10 x 2.5 = 25mm and 10 x 2.54 = 25.4mm so you could probably detect the difference with a standard metal metric ruler calibrated in mm. At 15 pitches it is 37.5 and 38.1 mm for a difference of 0.6mm. OR if it is some pitch closest to 2mm (unlikely) it will be different.
2 of these](http://www.pololu.com/catalog/product/973) from Polulu may do, depending on how they butt, BUT you can buy 32 pin ones.
An old floppy disk drive cable is 2 x 17 pin.
A HDD parallel access cable is 40 pin
Various examples here
Mouser have a very nice parametric selector that includes cable assemblies BUT no 32 way flat cables listed. (Check for yourself just in case).
Best Answer
I know this is not the answer you're looking for, but I'd just splice the wire back together unless you have a reason to think the connector itself was part of the problem. Someone here may be able to tell you the general name of the connector, or maybe the exact style or even a part number, but you could spend a long time finding one, and possibly for more money than you think that small thing is worth.
Also, many connectors of that sort require a tool to insert the contacts into the plastic housing. You don't have one of those, so unless you get a connector with leads attached you still have a problem. And then, what connects to the other end? A different connector?
If it were me and I was forced to replace the connector I'd check out the appliance repair shops or someone junking a used machine. I'd also try to determine what happened to the wire to assure myself that it was not going to happen again after the repair.