Just a point, the existing telephone jacks, are telephone jacks and not internet jacks. The way my house (a 2013 Gardenia) is wired, it would be some what difficult to do what is being requested.
The router would need to be installed where all of the telephone lines go (in my case the garage, the low voltage box). I do not think mine has sufficient room.
The cable modem, would then need to be connected to this router. In this case the existing wall plate needs to be removed, a new wall plate with two plug connections, 1 for phone and 1 for internet installed.
If you are lucky, and 8 conductor CAT 5 or better cable was installed, the spare conductors could be used for the internet.
If you are lucky, there will be some spare cable in the low voltage box to get the spare conductors to the new panel for the internet connections.
ETC ETC ETC
IMHO, go wireless it in much easier.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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