Audiobus: Use your music apps together.
What is Audiobus? — Audiobus is an award-winning music app for iPhone and iPad which lets you use your other music apps together. Chain effects on your favourite synth, run the output of apps or Audio Units into an app like GarageBand or Loopy, or select a different audio interface output for each app. Route MIDI between apps — drive a synth from a MIDI sequencer, or add an arpeggiator to your MIDI keyboard — or sync with your external MIDI gear. And control your entire setup from a MIDI controller.
Download on the App StoreAudiobus is the app that makes the rest of your setup better.
Comments
Get one, but don’t spend more than that.
I think you'd get some latency with Bluetooth?
While recording, you could use an audio cord to hook up the speaker to the keyboard with no latency (that's what I do). While playback, it doesn't make matter if using a cord to physically connect (no latency) or wirelessly connect using Bluetooth (in this case, all the playback has uniform latency).
You could find a used Bose mini SoundLink on eBay or Amazon for your budget:
https://www.ebay.com/sch/i.html?_nkw=bose+mini+soundlink+ii
https://www.amazon.com/gp/offer-listing/B00WK47VEW/?condition=all
The best quality speakers are small but heavy with a metallic frame like the above one.
@MobileMusic yep, I use a cable most of the time, although some speakers have latency even with a cable. Happily the Bose soundlink 2 that you mention sounds great and has no latency with a cable. I love mine!
Be careful. My Bose Soundlink III has some latency even when using a wired connection. It seems there is a lot of processing going on which gives a small but noticeable delay. It’s great for listening (either wired or Bluetooth) but if you are sensitive to latency playing live it’s not so good.
Yeah, no latency with a cord but ironically, Bose mini SoundLink sounds WAY BETTER than its big brother SoundLink III SoundLink mini is so small, you can take it anywhere and even take calls on it. Sometimes, I do orchestra with it in my car.
Also, the 1st gen mini SoundLink has a bit more bass than 2nd gen mini SoundLink II. The 2nd gen has audible alerts about battery level, 10-hour battery as opposed to 7 hour in 1st gen, etc.
1st gen - https://www.amazon.com/Bose-SoundLink-Bluetooth-Discontinued-Manufacturer/dp/B00D5Q75RC