"Hired into" is slightly unusual, although there's nothing ungrammatical about it. English often allows a choice of pronouns.
You normally talk about someone being "hired for" a job in general, or "hired as" a particular job or position. Lexico has "hires someone for a job working a cash register" and "hiring him as his personal instructor".
An alternative might be:
Kate has been hired for the sales department for her ability to connect with people.
"For" is repeated, which is the only problem, and not a big problem.
It's common to talk about someone "being transferred into a department", "getting promoted into a job", and similar things, so "into" isn't weird. You may like "hired into" because it emphasises that someone is joining a team, and teams are important in business.
Or you could certainly write it as:
Kate has been hired for a position in the sales department for her ability to connect with people.
That is a bit longer though.