In my experience with Vanilla RTW, the best way to deal with the Egyptians, and possibly the only way, is to strangle them economically. Keep in mind that its probably best to ensure that you have no other conflicts going on, or at least nothing too major, you'll need to put most if not all of your resources to work against them to be succesful.
Before you do anything, you must establish a giant "Spy" network. I say "Spy" because you dont necessarily need it to be all spies but being mostly spies will obviously be a plus. The reason is you need to have eyes all over the egyptian territories (retarded areas in the desert are obviously not included, that would be pointless) so you know where they are strong, where they are weak, and what is important. The spies of course will also have their natural tendency to cause slight unrest, and force the Egyptians to at least lower tax rates if nothing else.
Now find out who their neighbors are and how they are dealing with one another. If they have allies, ignore them, if they have enemies, give them money in order to fuel their war effort so as to better fight your common foe. Those who are neutral with the Egyptians you can try bribing into fighting them, or at least declaring war and severing trade rights with them.
Now build the best ships and make a decent sized fleet big enough to blockade each of their ports. Naval trade is one of the best ways to make money, and cutting them off from it will help out greatly in the longrun. In order to further increase the levels of economic hardship upon them, you can use a little trick I started using in order to deal with high density population centers. Build a massive number of peasant units, just spam the hell out of them, and load them up on your boats and scatter them throughout Egyptian lands. Position them on undefended roadways with heavy trade traffic and in regions with high fertility levels. If they get attacked, have them run and come back later, if they die, you can just spam more. This will not only screw with their trade revenues even further, but will also force them to spread out their forces in order to deal with your pillaging peasants.
Now that they're economy is under siege, their forces distracted, and hopefully their main armies at war with nearby nations, you must strike. You want to be able to keep your forces mobile and able to strike anywhere as soon as the opportunity presents itself so your attacks will almost always be launched from your ships. As soon as you see a territory with a small garrison, land, besiege, exterminate, then demolish. By the time you've reached the Egyptians they likely already have a huge financial gain, so you need to be ruthless if you want to have any effect on them. Burn everything to the ground. When the city is bare, simply leave and look for the next assault location (Or if you wanna be cheap, gift the city back to them, then exterminate again, wash;rinse;repeat). You want to keep Egyptian losses to a minimum at this point so that their revenue gets closer to red while their troop upkeep stays up.
Once you've managed to gut their coastal cities (Keep in mind that the coastal cities are the prime settlements, largest populations, high tech infrastructure, massive trade revenue, higher than normal fertility, etc) you'll be ready to begin moving in-land to make the final blow. Land the majority of your stacks along the Nile Delta and assault their major settlements and the capital. Once you take these, all the Egyptians will have left are desert provinces and will be unable to field any decent forces (Neither in quality or quantity) against you, and will be, for all intents and purposes, eliminated.
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