Hi!
I have just finished that mission on masochistic and it really is quite hard. Here's what I did, or what I discovered while playing.
The first thing is getting your HW runnign in high gear. The goal is to be able to build each second turn a ~150BC worth ship, and still be spending some 20BC in another area: building research centers, market centers, or reseaching. It helps tremedously if you have a good production bonus tile (300%, or 2*100%) on your HW. Then you put 100% spending to social, purchase a factory on that tile, build the second and the third (spending just the exact amount of money, excess money going to research Planetary Improvements). Then you have two options: if you have good production bonuses on planet, you can:
- put in queue manufacturng capital, again at social spending 100%, while growing pop at 100% approval or
- if there are good planets close, start building ships immediately and save Manuf. cap. for another, better planet.
In that time all your ships (but Miner) should be moving out in FOW to explore surroundings and find planets (well, not necessary, you alrady know that map).
In my case I continued with Manuf. cap. and postponed building colony ships because the map was big, other races far away, and there was not much planets to take between us. So my initiall colonizer went to the center, and I held the production of them until I'd found more planets. The chances seemed to be good I'll find some close. Soon it became clear I have only 2 normals close, the rest was radiated.
You should colonize both normal planets and grow them well (colonize with 300+M pop, put in queue 3-4 factories, one morale building, two market centers, 2-3 research buildings, some market centers, leave 1 or 2 tiles free for a farm; set focus on social, decrease taxes to have planet at 100% approval and forget it for a while). You can also divert some resources from asteroids to help the planet grow quicker.
Also, while building infrastructure on your HW build 3-4 anomaly-explorer ships with 2-3 Ion engines, rest range extenders and set them to autoexplore. By exploring anomalies they will give you enough money to run serious deficit spening (-200 to -300BC per week) for more than a year. Create a cheap constructor and take those two closest resources for an early boost. Don't use more than 3 per a resource though, as DLs primary target are starbases, so be ready to lose them and keep playing without them for quite a long time. In mid game I've taken most resources I found (but influence), just for that small boost, and to give DLs more targets. Also try to get both treaties with Thalans. You'll probably need to wait until your econ and reasearch grow quite a bit. There are also two normal planets in the center of the map. I managed to grab the most central one with my starting colonizer, and it was NEVER attacked by anyone.
Better econ buildings are also a MUST - try to get a Trade center ASAP. It's cheap, and each will boost your income on a planet by 15%. Lately you can replace them by research buildings or "wonders", but in the early phase they'll help you became financially independant. Better econ buildings are more important than better factories or research buildings, as they'll provide you money to actually run them.
Also keep pop at most planets at 100% approval until you'll almost run out of money, or most of planets will already grow to ~6B. When you run out of room to grow pop, it doesn't matter how much you tax them, until the approval is above 20% (below that you start losing them). If you want more pop (like you've built a farm on most planets), you just decrease taxes. The only real danger when there are no elections is invasions. Someone using Information Warfare will take your whole planet with the pop almost intact.
After about a year you'll see first DLs ships. That't he sign DLs' troop transports will come soon. And those are the real danger in the early game. Create a network of scanner ships (cargo hull, 2 Ion engines, 1 laser-5, rest sensors), and hold them 5-10 squares away from your planets in direction from where the DL ships are coming (left corner of the map). With Warp engine bonus they'll be able to spot and intercept every transport DLs will send. Be ready to replace them in case DLs' ships will try to kill them, or you'll lose them by those transports (in DA the attacker rolls from 0!). They are quite low in DLs' targer list, but anyway. If they'll target one, just send it to the farthest corner or the universe it can reach, and write it off. In my game some even survived that chase. DLs' pursuers probably got bored.

When your sensor net is ready (you have ~2 per outer planets, 1 for inner), grab those closet radiation planets (IIRC I had 3, and another one close to Thalans), and build them to 100%. In the mean time research all trade tech, send trade ships to Thalans, and build a Galactic Privateer "wonder". It will protect your trade lanes from being destroyed by DLs, and will aso keep them busy for some time, when they'll try to destroy them.
Now your economy is rolling in high gear and you're good to deal with DLs. As first research most ground battle tech and build a Tir-quan. When you 'll have that, the DLs' trop tranports will be just a minor nuisance (killing ~1.5B of your pop with each invasion). If you'll build Aphrodisiacs you'll replace lost pop in 5 turns.
The second goal is to kill those DLs ships. To find them build Eyes of the Universe first. To kill them: my best bet after finishing that mission is not fleets of small/medium ships, but single most powerfull ship you can get with the cost and tech. I suggest using large hull with one Warp engine, rest filled with weapons. The best early bang for the space gives you Harpoons, and very soon after them the Psionic missiles from that branch and Evil alignment. But in the late mid-game nothing beats Psionic beam. So research accordingly, build two-three ships, NEVER put them in a fleet, and go hunting DLs. The fregates are most dangerous, as they usually have 32 missile defense and 22 hit points, so your missile attack needs to be ~100 to have ~50% chance to kill it with first shoot. It will help if you immediatelly before you start attackign DLs in one turn build full starbase on that weapons resource that's about 3 sectors to your south. OFC keep one of your large ships in defense of it.

Also don't bother with HPs of your combat ships. Just make sure you attack first, and if your ships kills the target in first shoot, it will survive regardless of damage it will take.
After you've taken out all DLs' ships in space, it's time to take also their planet. This time you may consider using a fleet of two of those attackers, as in orbit of the DLs' home planet is a nasty cruiser with 50 beam defense and 50 HPs. You'll probably lose one of your ships. Just make sure you'll invade with 2B troops immediately after you clear the orbit, and DL's are no more.
Clearing the Korath after that will be a walk in a park. Im my game they went broke in first 5 turns, and never really recovered.
BTW don't bother with spying. On Korath it's a waste of money, on DLs it's pointless. The main DL's problem is MONEY, not production. Even when I had all their factories blocked, they still pumped out 80 MP from their unblockable capital.
Hope that helps.
BR, Iztok