Choice of Wire Gauge for 5v 26A LED Power Line
These are LEDs that have an integrated processor. So unlike a typical LED, controlling the voltage, instead of the current, to each LED is the primary design goal. As such, the issue of wire gauge becomes a question of voltage drop in the wire due to the current passing through the wire.

The LED spec sheet does not specify a minimum voltage but instead lists a typical voltage of 5.0 and a maximum of 5.5. For the purpose of this analysis, we will assume a minimum voltage of 4.75 volts. The maximum current per chip is 26.5 mA. Assuming 60 LED chips per meter, a 60 cm strip would draw 36 * 26.5 mA or ~ 1 amp.

To provide no less than 4.75 volts per strip, the wiring must have no more than 0.25 volts / 1 amp or 0.25 ohms. The longest wire length is 10 meters but there will be twice that in total wire length to account for the positive and negative wires. This gives us a resistance budget of 0.25 ohms / 20 meters or 0.0125 ohms per meter. Consulting a copper wire chart, 15 ga wire or larger would be suitable.

By following the above math, you can calculate the minimum wire gauge for any number of LEDs per wire run.

Given the large total current of the supply, it would be wise to fuse the various wire runs in no more than 5 amps per segment. The fuses should be located at the supply connector.