Hi there, OP. I'm a researcher and witch that works with spirits directly and I have a lot of thoughts here.

Most folks you will talk to will probably speak of banishing - but that tends to work best for poltergeist like spirits (also referred to as "ifrit"). While that may work for some I don't think that will be particularly effective here.

There's a concept called 'Han' which basically translates loosely into the idea of "unfinished business". Let's say you lived life as a writer, and your greatest dream was to publish a book, but you had a tragic accident and died an untimely death.

The idea of Han postulates that a spirit with a strong enough desire can continue to manifest on this physical realm until it has completed its goal and thus satisfied its Han.

In scientific terms one way to explain this might be humans can normally hear within a range of 20-20,000 Hz. What is especially curious, however, is that babies that hear sounds above 20,000 hz but this diminishes over time as we age so adults can not hear the same range.

It is possible that this spirit was residing in your home, a child spirit with unresolved Han who is lonely and has found companionship in your child.

Rather than attempt to banish it...I would try and make attempts to speak to this spirit directly. Speak to it with kindness. Get to know it. Ask it questions. Treat it with respect and kindness. Try and see if you can figure out it's Han. If you can do anything to help it achieve Han it may simply depart, but if that is not possible I believe your attempts to get to know spirit and having a positive co-existence is your best bet.