Ok, I'm guessing it's a RGB strip? Do you know if it's common anode or common cathode? It might be in the datasheet. Basically, are all the plusses form the leds connected together and the minuses go to the controller per color. Or is it the other way around? Most likely the connector has 4 pins. One of them is connected to all the leds, the other three are connected per color. You need to switch the common one. (Or all three of the other ones, but then you need 3 MOSFETs instead of one). The only downside is, if the common is the plus (which it most likely is) you need a p channel mosfet, they are a bit more expensive and less efficient. But should still produce a drop in power that shouldn't be visible.

Sorry, I had to work until 23.00 today. So i didn't have time for the schematic. But I looked at your schematic a bit better. You used a N-channel mosfet. There is a bit of nuance there. But it boils down to: The mosfet needs to be in the negative lead. So 24v+>led>mosfet>24v-. Using N-channel is the positive rail is possible, but not without extra drivers/boostcircuitry. But using your schematic you van just remove the MOSFET all together. A MOSFET, solid-state relay, spdt (or any non solid-state relay), bjt, and a bunch of other devices are basically switches. Sometimes you need to use a MOSFET to switch a relay, but the other way around is almost never the case. And that is what your doing in that schematich. Just remove the MOSFET and switch the positive rail with the SSR only. That should work just fine. I'm guessing the dimming effect is caused by the wrong usage of the N-channel MOSFET. But as some said before. SSR's, mosfets and a whole host of alternatieves range in their "on resistance" (in other words, the amount of voltage that drops over the switching device, resulting in the dimming of the leds) form 1 miliohm to over an ohm. On 240v that doesn't matter to much. Because 1v drop is still 239v. Or not even .5%. but 24v becomes 23v. Which is over 4%.

If I get a second tomorrow I'll draw the schematic! (I'm in industrial automation. So work tends to get a bit crazy when machines won't work as they are supposed to do :P)