Gussuri (ぐっすり) means sound asleep: ぐっすり ねました is “I slept really well”.
No dreams, no waking up at three in the morning, nothing until the alarm. You open your eyes and feel brand new. That night was ぐっすり.
What it means
ぐっすり describes sleep that is deep and unbroken. It is not really a sound: it describes how well you slept. Unlike most sound words, it is not a doubled pair. The small っ gives it its weight.
How to use it
Put it right before the verb: ぐっすり ねます, ぐっすり ねました, ぐっすり ねて います. It almost always goes with ねます, “sleep”.
おはよう ございます。 よく ねましたか。
Good morning. Did you sleep well?
はい、ぐっすり ねました！
Yes, I slept like a log!
よかったですね。
I'm glad.
Close cousins
すやすや is the quiet breathing of someone sleeping peacefully, often a baby or a pet. ぐうぐう is loud sleep, with snoring. うとうと is the opposite of ぐっすり: dozing off, nodding on the train or in a meeting.
In manga and anime
A sleeping character gets a bubble from their nose, and beside them ぐうぐう or すやすや. ぐっすり is more often heard the next morning, when someone says how well they slept.
Can you use it?
Yes, anywhere, even politely. Asking a guest ぐっすり ねましたか in the morning (“did you sleep well?”) is a kind thing to do.