Hi Karen, here's a few pointers, hope they are of some help.
When removing the old battery spring -
Wear glasses for eye protection.
Give the iron at least 6 minutes to heat up (and because its brand new, try to melt some solder onto the tip after a couple of minutes or so to protect the new tip from oxidisation.)
Before touching the iron to the solder joint to remove the battery spring, wipe the tip clean on an old wet rag and then tin it with solder, the tip should look like chrome with the fresh solder on it. If you have waited 20 seconds since you last tinned the tip, the solder on the tip will go gray, and you will need to re-tin it again. Having a clean tip with fresh solder is a necessity, don't worry about "wasting" solder during tinning, it is cheap enough, and it has to be done or the solder job will not be good.
It should only take around 5 seconds or maybe a few more of the soldering iron tip touching the solder on the old spring to fully melt the solder in order to remove the old spring.
To fit the new battery spring -
Place the new spring in position and tape it down with masking tape so it can't move.
Clean and tin the tip of the iron, hold it at an angle and place it so that you get maximum contact area between the soldering tip, the spring plate and the switch terminal and apply solder at the same time, use enough solder to see it bond the switch terminal to the spring plate; the instant you see that bonding thing happen remove the iron from the work and let it cool. If you leave the iron in there too long, the solder will go grey and the joint will be degraded. It should take around 5 seconds to do the actual soldering together part. If its good, it will look shiny.
If the solder is dull and grey, remove the spring base, tin the iron and drag the excess solder from it, and have another go at refitting it. The biggest danger is taking too long to get the solder to flow, which could overheat the switch terminal and destroy the switch. Then you will have to order a new switch and solder that in as well. But I'm sure you'll be fine. After soldering in one new battery spring, the second one will be much much easier (on the other Woodvil). You can use isopropyl alcohol and a Q-tip to remove the flux left from the soldering process, or nail polish remover.