Why don't my CAT5e patch cables work?

If the switches and router do not have auto-sensing, then you will need a crossover cable. In contradiction of Harry's answer, this will need A wiring on one end of the cable and B on the other, as per this reference:-

Crossover Cable

Some applications may require a crossover cable. The most common use of a crossover cable occurs in wiring together two Hubs. A crossover cable “crosses over” Transmit and Receive Data. Pins 1 and 3 are crossed over, and Pins 2 and 6 are crossed over. To build a CROSSOVER cable, simply wire one side according to specification T-568B, and wire the other side according to T-568A.

As to why your A<->B cable didn't work, I can only conclude that either you did not wire the connectors correctly or there was a fault in the cable run itself.

You should test with all the devices in the same room: the indicator lights will tell you whether a connection is established.

My problem was caused by something seemingly obvious: I hadn't actually crimped the cables. I had read that this was probably going to take a few tries to get right, so I wrongly assumed "surely people aren't throwing away dozens of RJ45 jacks" which was incorrect.

For future readers, some things to check when asking yourself the question of the OP are the following:

• Have I actually crimped the cable? (this forces down the gold pins, through the plastic coating, to actually make contact with the copper)
• Do all of the wires reach the end of the jack? (these should all be the same length)
• Are my cables in the correct order? (I used T568B -> T568B for straight-through cables)
• Are there any device faults? (check the device you are testing with works with other off-the-shelf cables)
• Are there any cable faults? (check the cable you are using isn't faulty using a cable tester, I got mine for ~£8 on eBay)
• Are there any sharp twists/bends in the cabling? (this can slow down your connection, if not break it completely)

EDIT: This is based on my few days of experience; please see this answer for a highly detailed and more accurate explanation.

You cannot mix A and B standards on one cable. Patch both sides of one cable with the T-568A Standard...

Unfortunately i cannot tell what you did wrong at your first attemt :-( Possibly you mixed up directions?

Both sides need to start with Green/White on the left side