Well, I sure learned some stuff about 'Busy Lizzies'...I have NEVER heard this name for impatiens. Always learning something new. In addition, I just learned that impatiens have a serious disease where we are supposed to rip them out and dispose of them in a proper facility to not spread the spores of this Balsam Blight. An amazing quirk in our nursery hothouses. I had no idea.
Yours is indoors in an office and it sounds as if you've had it there for a few seasons? Has it ever flowered? Is it in potting soil out of a bag or does it look like garden soil?
These 'droplets'...hummmm. I am going to just have to use my critical thinking as my library is not here and certainly could not find a direct answer off this internet. These droplets must be carbohydrates excreted via stomata. I am thinking this plant is using this sugary liquid as a 'trap crop' so insects don't ruin the flowers. The flowers are self pollinating and insects would only waste their time. The insects can be satisfied by easy to harvest sugary globules and leave the flowers alone. Interesting!
The main import here is that impatiens to include the hardier New Guinea type have gotten some gnarly disease and we are supposed to NOT purchase or plant them, get rid of them and any soil that has been connected with Impatiens to include their dropped leaves, again, to try to suppress this Balsam Blight. Your plant is actually a rarity...or from a very naughty nursery trying to unload their impatiens. Impatiens was a multi million dollar industry! And this blight is costing lots of millions of bucks.
I had no idea! Thank you for this question!! I wouldn't worry about these 'secretions'. Not a huge population of insects to attract in an office. Cut flowers off as soon as they start aging. I always cut flowers and flower buds off any plant that needs a boost, without waiting to see the pretties. Pinch back the apical or terminal buds to put more energy into the main plant. Do not use too much fertilizer and make sure the NPK formulation is lower N than the P and K percentage numbers. Don't worry if you seen any insects who are attracted by the 'sweets'.
Make sure to keep this plant and its soil inside, never outside and when you dispose of it put it in a plastic bag to take to the dump who should know how to handle problem organics.
I had no idea this had happened with Impatiens. Major bummer. Not many annuals at all that flower in shade like Impatiens! They are telling us to find substitutes so that we can get rid of this blight. Like begonias... You do know that impatiens is classified as an annual. Very short lived. Don't feel guilty when this plant passes on. There are far better indoor plants. In fact, I've never planted impatiens for an indoor application. But I just found out that this plant needs to be cleaned up and gone fallow to help reduce the danger of this disease being perpetuated. So we can once again plant impatiens for color, for flowers...in the shade out of doors!