Hi Byrdie. You said the first tree died of a disease..Do you know what type of disease? Was it insects by chance?

This small, white sticky thing sounds like a bug..It could be mealy (white) Scale (brown) or citrus scale.(white) http://en.wikipedia.org/wiki/Scale_insect
http://en.wikipedia.org/wiki/Mealybug

Check the sites above to ID pest.
In the meantime, for scale, I use 1 capful of Fish Emulsion per 16oz mister of water.
You should hand-remove as many insects as possible. Squish.
Mealybug is a little more difficult dealing with,(IMO) Most people wipe them off using rubbing alcohol w/cotton ball.. I'd use Fish Emulsion for either pests..
Fish Emulsion has a fishy odor that lasts a couple days indoors..but they also sell an odor-less type. Still, if a couple days fishy odor killed these pests, wouldn't it be worth it??
One thorough spray should rid them.
Scale secrete honeydew (sticky substance) which is probably what you're seeing, though I've never known it to be white.
On the other hand, mealybug secrete a waxy substance, but I'm unsure of its color..

Insects dote on wet soil and dry air. Improve their environment. Run a humidifer if your house is dry, mist leaves, or set on a tray w/pebbles.
Be sure soil dries before watering. Proper watering is really important.
There are stronger chemical insecticides, (I don't use them,) so this is up to you. Diazinon is one, but several applications may be needed..Good luck, Toni

First, you must take care of the insect problem. I would first take the tree outside and hose down the undersides of the leaves to get rid of any insects and their eggs. If you can't bring it outside, take it into the shower and use the shower hose on it. Then, I would use some insecticidal soap on it (which has no bad odors). Once this is done you can work on improving its indoor quality of life for the winter. Do you have a place for your lemon that is really cool (enclosed porch, garage, etc.)? I keep my citrus in a garage where temps are between 60 and 40 F. Garages faces due south and the top half of the garage doors are glass so they do get a fair amount of sun in the winter. At these temperatures, I water very seldomly, rolling them out on to the driveway on mild days and hosing them down (maybe once a month). I will mist the leaves occasionally. My collection includes Satsuma mandarin, Valencia Orange, Washington Navel Orange, Merer' Lemon, Leureka Lemon, Citron, two varieties of kumquats, Calamondins, limequats, and a Persian Lime. I only have one citrus plant in the house over winter, a Kaffir lime. Keeping citrus cool reduces/eliminates any bug problems in the winter (citrus are vulnerable to insect problems under warmer indoor temps). I roll the citrus out in early April and they summer on the wall of the garage, in full sun, until the end of November. Although I water sparingly in the winter, summer is a different story. They get hosed down nearly every day--depending on temps and rainfall. In between seasons, I let the soil dry moderately between waterings. Right now the Washington Navel is in bloom so it must like the conditions. Good luck and Happy New Year!