You can export your playlists in m3u and then have Kodi scan your music files and whatever folder your exported playlists are in. That's probably the easiest most reliable way. I had this set up a while ago and it works fine.
You could also do as phred suggests and use the hit-or-miss UPnP/DLNA plugin, which will serve up your library as it exists with all of your library data. That's a valid workable solution, but if you're just interested in listening to your music and playlists on Kodi it seems like slight overkill for what you're looking to accomplish. Plus you'll have to transcode to WAV to level volumes across tracks and crossfade and things like that, which I think Kodi will do natively.
I have a Windows 10 HTPC hooked up to each of my media centers, so that's my actual suggestion. You don't need to go that far, though. There are cheap Windows 10 boxes and even sticks (under $100) where you could install Kodi and MusicBee and other low-intensity programs (i.e. not things like Photoshop or office) and stream whatever. That's, also, probably overkill since you already have a streaming box, but something to keep in mind when it's time to upgrade.