Electronic – smd resistor and regular through hole resistor

Ideally the LEDs should be driven by a constant-current circuit. This will maintain a constant brightness and color as the battery drains, or as the LEDs heat up or cool down.

But the real world isn't ideal, so you can often get away by just using resistors. Yes, you should definitely use them. The resistors are there to limit the current through the LED and keep them from overheating and burning up. A 9V battery has a fairly high internal resistance, so you may be able to get away with two in series and no resistor, but it will be unreliable (changing to a different brand of battery could be enough to blow out the LEDs, etc.)

For the worst case of two white LEDs in series running at 20mA, the lowest forward voltage shown in your link is 3.2V, so you would have (9 - 6.4)/.02 = 130 ohms. The current is low, so a 1/4 watt resistor will be fine. Select the closest value to this you can find. Running at 20mA the LEDs will be pretty bright and this is a benefit: as the battery drains or the LED forward voltage changes, the apparent brightness probably won't change that much. Human vision is more sensitive to dim lights and it's harder to tell that a bright light has changed 10% than a dim light has changed 10%.

You'll need an 8-WFDN adapter, which is a different footprint than the SOP8/SSOP8/TSSOP8/SOIC8 footprints on the adapter you linked. Probably not easy to find - it's not a footprint commonly used by hobbyists. If you do find one, it may well end up costing more than buying similar components in another package.

If you think you can solder it (which might be very tricky without a reflow oven) then you could design your own adapter using EAGLE or similar - the footprint should be available for an existing part, and all you need to do is hook up the relevent pins to rows of 0.1" holes. There's companies in China which will do 5x5cm PCBs for about $10 (I've never used iteadstudio, but I've read positive reviews on the web). SMD to DIP adapters really are among the simplest boards you can design.

Your last option would be to solder leads to it "dead bug" style. It is an absolutely tiny chip though. The smallest SMD components I've soldered were MSOP packages, which are 50% wider than the WFDN package and also have leads (much easier to solder). Even then I don't think I'd have been able to do it without a microscope. I suppose if you've got the chips anyway it's worth a try, but don't be surprised if you destroy them all attempting to solder them.