Integrate Broadlink Mini to SmartThings app?

There’s no simple or easy way. And in all cases you will need one additional device to act as a “man in the middle.“

If you have a strong technical background and you would like to see what other community members have done (and you have a raspberry pi and are comfortable programming it), talk to the people in the following thread:

[RELEASE] Generic TV Remote Device Handler using Broadlink RM Pro and Mini with Alexa Support

But to be honest, it’s easier to just get a harmony home hub, which does have an official integration.