Bike has set for 4 years without starting it.
If the bike has sat for this long, usually the fuel starts to oxidize and decay, causing sludge and varnish deposits to form, which can start to plug passages in the carb body. It is usually necessary to dismantle the carbs, and give them a thorough cleaning. You can buy "Carb Cleaner" in various forms. I personally would dismantle the carbs completely, and completely immerse them in a vat of cleaner. To do this, you will have to remove ALL the rubber parts. You also need to dismantle as much of the carbs as you can. This will allow you to inspect all the parts, and see if anything needs replacing. Certainly, you DO need a new needle, and a new float valve. You can get them from Suzuki, or other on-line sources, such as "Bike Bandit".
You mentioned that the bike flooded that crankcase with fuel. This might indicate that the vacuum petcock may need to be replaced.
You picture also shows oil has collected on one of the intake valves. Are you sure this is oil? Could it be oxidized fuel from when the carbs leaked fuel into the crankcase? It is likley this is the intake that had the bad float needle valve, and this is left over oxidized fuel. You need to verify this. If it is really oil, then if has to be leaking down past the valve seal, and since the bike has not run in 4 years, it is not likley that there has been enough oil in the top end to leak down into the intake valve.
Anyway, here is a start:
1. Drain ALL of the old fuel from the tank. I would remove the petcock and make sure you drain it compeletely
2. dismantle and clean the carbs, replacing any suspect parts. Check all the o-rings, etc.
3. Likely new Air filter
4. fresh gasoline, likley new spark plugs, make sure the battery is charged and in good condition, and then give it another go...