Google can be your friend in such matters. Googling 'tnc connector' produces many hits and pictures. e.g.:-
I do not know this router, but suspect a special crimping tool (or press) was used, for assembling the socket and co-ax cable, when the circuit board was made.
The parts you show, relate to the co-ax cable screen. On your second picture (one marked small hole) should there be a metal pin on the inner cable? I cannot see if it is present.
Have you asked Linksys if they have offer an exchange /repair service?
This does not seem to me mentioned on their website support pages though.
Here in the UK, these routers are around £23 new. Vastly less if used, and possibly later/better models of such. (e.g. Ebay item 120925581037)
At these prices they are really throwaway items. Jefro is correct to advise purchasing a replacement as you will probably spend more in time and/or money attempting to repair without guarantee of success.
If you replace with similar unit, suggest keeping old one for spares.
Please advise back on progress.
As a matter of interest, where are you based. I am in Hammersmith, West London.
Good Luck - Keep us posted.