You f---ed up by not specifying two cables in seperate conduits, (not that you knew that) but the electrical subcontractor also f---ed up by not installing the conduit with mousing line through it to pull in more cables. Usually, you can only mouse a cable through if there is no more than one 90 deg swept bend in the conduit, (or swept bends totalling 90 deg), otherwise there's too much friction, so if neither building has a cellar that would have required an access pit and cover. It only has to be big enough to get your arm down, with enough room for a straight pull on the mousing line.
How to fix it? Well as its a dwelling for disabled elderly, a high reliability telephone connection that will work if there's a power cut is a must for emergency use. You may in the future need to install an elderly alert system, which *WILL* need a wired connection to its base statiion and has limited guaranteed wireless range. A DECT cordless isn't reliable enough, especially at extended range, nor are VOIP systems reliable enough, even if you put the DECT basestation or the VOIP adapter, router and your ADSL modem/router on UPSes.
If you are willing to settle for low performance and reduced security on the network, you may be to get away with a pair of 2.4Ghz WiFi access points with directional antennae acting as a wireless bridge between two wired network, to free up the Cat6 for the phone line. However unless the annex is an outbuilding far from the main building, iit will probably be cheaper and simpler to run an appropriately rated four wire exterior phone cable. Use the armoured direct burial type if it has to go under any paths etc. or run low down on exterior walls, or run on the outside of boundary walls etc. or regular exterior cable if you can run it so it isn't likely to get damaged.