If you imagine thr keyboard as a matrix of X rows and Y columns.
When a key is pressed all it is doing is shorting one row to one column.
So on the little circuit board there is a long row of contact points connected to all the flimsy membrane plastic stuff inside the keyboard.
All you have to figure out is whih of the contact points is a row and which is a column.
They are usualy organized so all the row contact points are next to each other and all the column pints are next to each other, but not allways.
If you look at all the litle tracks on the membrane plastic stuff you should be able to trace them back to the circuit board and figure out which two circuit point on the board any given button connects when pressed.
You can then soldere an ordinary switch to those 2 points and presto
your new switch will do exactly the same as pressing the original keyboard switch.
It's a little tedious to do all the tracing but you could also just hook the keyboard up to a PC and try to connect random pairs of contact points on the circuit board until you find some that give you the characters you want in your "mini keypad"