I wanted to write a comment, but it got quite long-winded... The formatting of an answer will help, and these things should get you started to find out what's going on. I'm happy to revise as more information comes to light.
The First thing you want to do is understand:
- Where the wires run
- If they are functional
If you're lucky, and all the wires are "functional", then you can easily figure out where the runs go with just a computer and a switch. By "functional", I'm meaning that an Ethernet connection can be established - blinky "link" LED on the switch, computer reporting a connection.
You'll want to make sure that the connection is at the expected speed... If you're using a gigabit switch, and a computer that is capable of a gigabit, then Windows (or
ethtool should report a 1000Mbps link. If it does not, then this can indicate problems.
If the hardware is capable of 1Gbps, but the link is reported as 100Mbps, or appears to keep disconnecting, then you'll need to at least invest in some basic tools, or call in a pro.
It looks like you've identified where the runs go, but it would be good to prove out those connections that currently "don't get internet" - don't look for internet connectivity at first, look for Ethernet connectivity. Then for communication with your router or another PC - use
ping -t and check for packet loss.
From your diagram, there should be 5x sockets in the store room. You'll want to put your router in the main hall (close to the "fibre" point), and then use a switch in the store room to connect to the rest of the house.
You can easily get your hands on a "Cable Tester" that will test "CAT-5e" or "RJ45"... search eBay, I can get one for £3.70 GBP (free postage)... there's no need to pay more.
With this you can connect one piece to each end of the run, and simply ensure that the termination is correct - the LEDs should run 1 through 8 on both ends, otherwise there is a bad link / mismatch.
More (much more) expensive tools will be able to provide you with information on the quality of the run - crosstalk, bandwidth, shorts, length, etc... I'm not advising you to go for a tool like this... Call a pro if you need to.
I'm not familiar with the terminals you've shown... I can only presume that the RJ45 socket is in a separate piece of the faceplate... and I also have to presume that by removing it you've pulled the wires from the punchdown block.
The 'keystone' terminals are fairly popular, shown below with a "punchdown tool" in action.
To put things back together, you'll need a "Punchdown tool" - again available on eBay for around £5 GBP. (You could even get a kit with cable tester, crimps, punchdown tool and a few other bits for ~£10 GBP!)
Termination is actually quite important - mainly, don't untwist the pairs too far and don't leave wire past the terminal.
As already mentioned in the comments, don't talk to electricians. They are not equipped to deal with Ethernet - or even phone lines in my opinion.