I came across the word keenly from a book which said "This world was being watched keenly and closely by intelligence etc."

keenly: with great sensitivity and intelligence
Synonyms: perceptively, sharply, shrewdly.

When I come across a new word I try to find others ways in creating sentences and phrases for that word.

I am having trouble figuring out when I can use this word in a sentence so far I got these sentences which I dont know if I am using it correctly.

She keenly applied her eye-liner before the show
He glances keenly over at his partner
I am keenly aware of my own shortcomings

I would love to hear more sentences and phrases.

Hello Henry and welcome to the site.

I can only give my opinion insofar as British English is concerned. However, here we would also expect "keen" to convey the sense of "with enthusiasm or particular affection".

"I am keen on her" = I like her a lot.

I would tend to avoid the adverb unless it conveyed that sense - which the exemplar sentence does. It is, however, rather more difficult to understand how eye-liner could be applied keenly - carefully, maybe.