Wi-Fi Problem

> [...] The router is in one downstairs room, [...]

   With my weak psychic powers, "the router" is not a very useful
description of anything.

> [...] I couldn't find a Wi-Fi unit with an ethernet input, [...]

   Where did you look for what?  Practically any wireless router can be
configured as a wireless access point, which seems to be what you want.
There also exist purpose-built WAPs.

> 1. Use the Cat5e cable and plug it into a Wi-Fi broadcast unit (they
> surely must exist?), [...]

   Wireless access point.

> [...] and then put a USB Network adapter into the Sky box.

   I know nothing about your (unspecified) "the Sky box", but, before I
tried that, I would look for documentation which says that you can use
any kind of "USB Network adapter" (wireless?) with it.

> 2. Use the Cat5e cable and plug it into a W-Fi broadcast unit that has
> an ethernet pass-through or an ethernet output. Do they exist?

   Wireless access point.

> 3. Interrupt the DSL router feed with a modem, have one output going
> to the existing router, and then add a second router in the dead zone.

   You lost me. How does this (unspecified) "the DSL router" compare
with your (unspecified) "the router" or your (unspecified) "the existing
router"?

   It sounds as if you might want to connect two routers to one modem,
which wouldn't work.

   "solution" 2 sounds plausible.  All you'd need would be a wireless
access point (or a wireless router which is configured as a WAP).

   Being conservative, I'd try to find some cheap/used router in a local
thrift/junk store or friend's junk pile, configure it as a WAP, and run
the experiment.  Then, if the proof-of-concept works, but better
performance is needed, look for something better to use as a WAP.

   General info: Terms like "input" and "output" don't really apply to
this kind of networking; all the paths and ports are bidirectional.

   For practically any reasonably recent Netgear Rxxxx model, visit
http://netgear.com/support , put in the model number, and look for
Documentation.  Get the User Manual (at least).  Read.  Look for a topic
like "Use the Router as a WiFi Access Point Only" or simply "access
point".  For other makers' stuff, check the appropriate docs.  There
also exist general guides for router-as-WAP, such as:

      https://community.netgear.com/t5/x/x/m-p/1463500

That's written for a Netgear C6300-as-WAP, but the steps are about the
same for any other router (any make/model) which lacks a one-step WAP
option.  If the router which you're reconfiguring has a WAN/Internet
Ethernet port (unlike a Cxxxx), then leave it unconnected.