This is the 1383rd most frequent Polish word.
ręką
"Ręką" means "with (a/the) hand" in English.
In this sentence, 'ręką' refers to the use of one's hand to perform an action, indicating the instrumental case in Polish.
Dotykam stołu ręką.
I touch the table with my hand.
Here, 'ręką' is used in the instrumental case to specify the tool or means by which the action of lifting is performed.
Podniosłem książkę ręką.
I lifted the book with my hand.
In this context, 'ręką' emphasizes the instrumentality of the hand in writing the letter, again employing the instrumental case.