But, when a key is down, the resultant resistance will be 1/2 of whatever value is on a specific switch.
That's right, but only when pressing one key.
At 8 Key's are conected the MKx lines and the BRKx Lines together, and pulled up with 2 resistors.
Next 8 Key's the same and so on.
The Tx lines of the first key of every group are connected together.
The seconds key also.
Let's say the first key of the first group is pressed, also the first key from the second group.
The MKx and the BRKx Lines of both groups must be pulled down.
So You have to pull down 4 resistors ( MKx1, MKx2, BRKx1, BRKx2 ).
You'll loss about 0,6V about the diodes, and a non specific voltage by the used driver, depending by the current.
In sum it could be possible, that the scan input can not further detect a pressed key when the voltage don't come down to LOW level.
When 15k work fine without dissortion of the scanning signal all is fine.
When get a low edge steepness the values should be decreased.
But not to low, because the written issues.
A look into the datasheet of the scanning chips could help to figure out the voltage for a proper low level detection.