Dude, first, I would keep on it. The more you let it get gunked up, the harder it is to clean. First, make sure your stem full is really cashed. If it isn't, some debris can stay on the screen, especially around the little retaining ring. I would use either ISO or something like 151 or Everclear. I use a small travel bottle with a cap and let the stem soak, tip down for a while (how long depends on how dirty). Sometimes, I'll shake the bottle, but if there is a lot of debris in the ISO from previous cleanings, it can lodge in the screen. You may need to use something like a Q-tip to gently dislodge any stubborn debris, both in the stem and the tip. Just be real careful around the screen area, you don't want to remove the screen or dislodge it. You should not have to replace the screen for a long time if you keep up with your cleaning and are careful not to poke it out. However, if you should poke it out, it is pretty easy to put back in.
The screen replacement kit is a nice thing to have, but not necessary. It includes some tools that make it much easier when putting in the screens and clips.