My general process, assuming no one's doing anything unsafe:
Get the audience out (including unnecessary staff). Extra people are not good for drama-seekers, and not good for people who are panicked and overstimulated. Only one person should talk to the patient. Don't let people stand/loom over the patient or make comments. Get down to his/her level yourself.
Introduce yourself, ask permission to do a quick physical check. Wait to get OK. Listen to lungs/heart and check O2 sats. If everything's OK, verbalize findings "excellent, just need one finger, I'm checking your oxygen. Your brain and body are getting plenty of oxygen. Now I'll check your lungs. You'll feel the stethoscope here. Very good. I know you're short of breath but don't worry, your lungs sound good. It's OK to slow down your breathing now, I'll keep an eye on you." (If nothing else, this reassures admin)
Give reassurance and two safe choices that you can offer: "I'm going to be with you until you feel better. Do you want to rest here, or would you like to get some air in the courtyard?" No other questions, no longer statements. WAIT for answers longer than you think, you might sit quietly for a what feels like forever.
Other things that I try to remember
You don't have to fix anything. If everyone's safe, it's OK. Panic passes, no matter what you do, so just being there is fine.
Sometimes I tell kids this! "This terrible feeling is like a big wave, it can knock you over for a minute, but then little by little it rolls away and leaves you safe on the beach."
When they're out of the full blown panic, I'll try techniques folks have mentioned above. If I know they're working on coping skills, I'll offer their preferred strategies: "I'd like to help you use a coping skill, should we try music or breathing?"
Some kids do better with just decreased stimuli. I have a room in my office with.a recliner and a dim light - I will say, "would it help if you had a quiet, private place to regroup? Yes? Here's a glass of water. I'm right outside if you need me. I'll check on you in 10 minutes." (of course this is presuming no self-harm behavior, and that I can discreetly keep an eye on them). Sometimes I'll ask where their anxiety is (1-10) before and after.
No one teaches us this stuff in nursing school! I just try to project kindness, confidence, and a neutral tone. Practice helps. You can do it 🙂