One issue that I see is that if it's a flying dragon maybe being 2 miles long strains the imagination somewhat. Also, there are not many materials sturdy enough for such a large object to be made. For instance bones are pretty close to the limit with something like an elephant. If it's 2 miles long, what are its legs made of? How does it not sink into the ground?
With regards to storing the radiation, the problem is shielding. Alpha and beta are trivial to shield against. Even a paper sheet will do. Gamma is a lot harder. To stop those, you need a lot of matter, and it has to be something dense, like lead. Now you have the issue of this thing moving around with the weight of the reactor and the shield. In fact, the closest thing to what you describe is a modern aircraft carrier, which is pretty much only possible on water. Maybe it should be a water dragon.
Second, the part about the breath weapon. First of all, the dragon need not breathe, but can simply store quantities of radioactive fuel somewhere like above its eyes, and have dangerous "heat vision". You can make the aperture out of lead bearing proteins or whatever (actually, it really has to be a solid lead, any protein with lead co-factor would have way too much gamma-transparent non-lead, so you would need A LOT of the shielding, like dozens of meters thick probably). Second, in reality a lot of the danger of nuclear bombs is not the explosion. That's gone in an instant. What happens though is a lot of radioactive dust is generated. You breathe/drink/eat this dust and then it shines radiation on the inside of your lungs, which is how you get cancer. The breath sounds like it has radioactive dust in it. So how would it avoid breathing this dust itself? Spitting a solution of radioactive dust would be more believable. Also, as others point out, it wouldn't be an instant effect. So perhaps it makes more sense for the dragon to breathe superhot steam from the reactor cooler, or plasma (for instance, directly from the reactor if it's a fusion dragon).
Fusion dragon would also be nicer because the fuel is just hydrogen, easier to find than Uranium ore. Also, ore actually has a tiny amount of active isotope. Tiny amounts of it must be filtered from huge quantities of ore. This is done with a centrifuge, because a slight difference in mass is really the only way to distinguish the two isotopes, they are otherwise chemically identical. So your dragon must eat A LOT of these rocks (how???) to get enough fuel, even if he located a nice exposed uranium vein. And it won't be easy for his cells to chemically enrich it, he'd need a centrifuge organ (???) or something equivalent. It's also pretty incredible that you could biochemically enrich uranium, because it's so hard for living cells to distinguish isotopes that whole experimental methods are based around it. I'm not aware of a single conclusive case where a biological system could meaningfully distinguish isotopes.
I think however you always have the option of simply not dealing with any of this. Just declare that the dragon's biology is radically different and much less vulnerable to radiation. Maybe it doesn't have DNA but entirely different biochemistry. Maybe it has exceptional mutation tolerance. Maybe its immune system is really good at fighting cancer. Who knows, I mean, it's a 2 mile animal, it's clearly nothing like any living creature we know.
And lastly, the dragon's breath doesn't have to actually be deadly. A creature like this sounds like bad news, I wouldn't want to mess with it. So I bet a lot of information about its weapons is 2nd, 3rd and 4th hand. It's rumor and legend. Perhaps the dragon's breath doesn't instantly kill, but after he attacks the crops die and children are born malformed for decades, such that the town is eradicated. Over many retellings, this time aspect gets blurred, and people start thinking that the dragon just breathed once and everyone instantly turned into mutants. Not like many people would track him down to try and see for themselves.
