RJ45 Split, one part Ethernet other part telephone.
Those splitters work because Ethernet (non-Gigabit, at least) only uses two of the four pairs (pins 1,2 and 3,6) of the RJ45 connector and cabling. So they put the same pins of the second connection onto the unused pairs of the first (pins 4,5 and 7,8) and then restore them at the other end. Here's an example:



But normal wiring for telephones over structured cabling uses one or two of the centre pairs (pins 4,5 and maybe 3,6) so the same adaptors won't transfer the correct connections across. As 3 and 6 WILL be carried correctly (because they are also used for Ethernet) you can assume that it's 4 and 5 you're missing. As you can see from the picture above, 4 and 5 aren't connected at all on the splitter you'd use in cable sharing for Ethernet.

If you can't buy something suitable you'll need to make up some adaptors of your own that connect them correctly.

A quick search on Amazon found this adapter but it shows as unavailable, though I'm sure alternatives can't be too hard to find. You can see that it connects 1,2,3 and 6 for the network ("computer") and 4,5 for the phone.