Most people prune rhododendrons and azaleas after they bloom but before mid-June. This accomplishes 2 goals, it preserves the current seasons bloom, and insures time for the plant to produce flower buds for the next years bloom. Actually the best time to prune rhododendrons and azaleas is in March since this gives 2 extra months for regrowth.

If necessary, you can remove a great deal of material. When doing this it is best to prune in March to allow time for regrowth. It is a general rule to not remove over 1/3 of the leaf area each year. This should also leave adequate buds to provide plenty of bloom. Pruning is generally used to control unsatisfactory height or width of a plant. I don't prune very often and try to limit pruning to plants which have a shape that is unsatisfactory or dead branches. If I want to cut trusses for bouquets, I always cut the tallest flowers since this helps keep the plant within bounds.

Severe pruning is not uncommon with rhododendrons and azaleas. Sometimes a healthy plant can be cut to the ground and will come back, but this is very risky. Rhododendrons and azaleas have dormant buds beneath the bark, which sprout to form new growth after severe pruning. However, Richard Colbert reported that such attempts at Tyler Arboretum were only successful if the plant had enough sun light. Those in heavy shade frequently died. He recommend first opening up the shade by thinning the forest canopy. Then he recommends just removing some of the top to induce new growth at the base. Then when that new growth is established, the remainder of the top can be removed.

Here is a link that might be useful: How to grow rhododendrons and azaleas

While it may seem as if pruning now would be a no lose proposition, it would be far better to wait and then follow rhodyman's advice. Pruning even at this late date can stimulate growth. This growth would have no time to harden off before seriously cold weather and be winter killed. You'd be left with a smaller rhododnedron, but likely one with many unsightly bare patches.

What to do with huge old rhododendrons - they're usually rhododendron maximum - is a vexing problem when they've been planted too close to the house. Best approach is probably to trim back conservatively the first year to see how it responds, then more in the following two years if it regrows well.

Failure to set flower buds is usually the result of too little sun, but in the case of an old negleted plant like this one a soil test is probably a good idea. It's also not clear in what way some buds looked unhealthy. This would be worth investigating as well.