Please forgive the Telecom 101, but I think everyone might be misunderstand the way in which Unshielded Twisted Pair or U.T.P. cable works. The original ideas behind the need to reject noise in the minds of early Telecom Engineers was the rejection of the most fundamental noise creator that in many cases 60Hz AC which was on the same pole or at least many times crossed their cables at intersections.
The effect the twisting of the pairs has is that the induced noise is hopefully out of phase on each conductor, and when used to input a differential input on an Audio Amplifier the noise or induced 60Hz would be out of phase and cancel itself out thus producing no output on the other side of the amplifier. Cat 5e for Ethernet depends on the same cancelling out effect to a differential driver as this is how the layer 1 works since it uses a Manchester Encoding Technique. Various digital encoding schema also take advantage of this effect when they split data into an I & Q data stream as well as an I & Q inverted stream called I & Q NOT for logical NOT. None of these will have any effects on the RFI, and I also suspect that there could possibly exist coupling, but most better routers have optically isolated inputs to prevent DC Coupling or inadvertently forming ground loops in the systems interconnected by the router.
If the 61KHz is being radiated by the cabling them possibly upgrading to a CAT 6 with a grounded shield would prevent some retransmission if that is even happening. I suspect their is no potential traveling down the cables so I would advise against the expense of the CAT 6 cabling that would provide a shield. The 61KHz sounds interestingly like an "by-product" of switching SCR's that are common in UPS's. Is it at all possible that an UPS is generating the noise? If not then we are back to some kind of DC-DC Convertor that almost always uses a really cheaper method of DC to DC conversion that uses a DC to AC conversion using again SCR's then using a small step up or down common by winding the secondaries of transformers differently for the different voltages, and then full wave rectification to the desired DC Voltage. The SCR's commonly are gated or switched by a Pulse Width Modulated technique gating the SCR's with this higher frequency waveform, and again this is very common in UPS's but unlike the cheaper Cisco / Linksys junk the UPS's manufactures usually include very expensive filtering to eliminate all of the AC ripple common on the output of all power supplies.
I could be totally off base here ,but a few checks with either Frequency Selective Voltmeter or an O-Scope would be very helpful. The fact many people are seeing the same thing , and the frequency makes me suspect a power supply design that might be a common sub-assembly that all vendors OEM from an outside vendor which is using Switching SCR's. I would suggest looking at the cables even the Grounds like in the AC Power Cord.
The commercial 100KW UPS I worked on 30 years ago used 100KHz as the Switching Gate Frequency so I would be real interested in knowing the results if anyone can do the measurements with an O-Scope, and if you could do a storage scope you would see the noise is actually a series of Bursts on and off if it is the problem I was proposing someone investigate with a scope.