It's easier to work on this problem if you zoom in and get a closer look at the problem.
When you choose the tool you will get a scope-like cursor. If it is big enough to encompass the eye you can click otherwise you can click and drag to adjust the size.
Click and the tool will seek and destroy the red. Now if you put your cursor over the red-eye circle it turns into a hand. This allows you to move the circle and better align the removal area. If you hear a warning tone when you first click it means that the tool did not sense enough red to perform the removal. Try clicking a different part of the eye.
If the tool leaves some red fringe you can adjust the size of the removal area. Bring you cursor near the edge of the circle and it will change to a bar with outward point arrows. Click and drag to alter the size and shape of the removal area.
Lightroom's red-eye removal tool provides two sliders to let you fine tune the removal: Pupil Size and Darken. Sometimes when the redeye is removed the pupil will appear light or ghostly. Use the sliders to adjust the size and darkness of the pupil. This will enhance the effect and provide for a more natural looking eye.
Once you are happy with the result on one eye go ahead and start on the other.
So if it's not Halloween then let Lightroom help you get the red out!

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