Happy to help!
Although, bear in mind that many of the details will ultimately be "whatever you like", so bear in mind that there's no one objectively correct way to resolve your fire spirit's engulf attack

So, the groundwork:
First of all, it's my personal belief that the Fatigue I status probably shouldn't be applied, as per pg 223 fire engulf instead applies the burning status... and instead Air Earth and Water spirits are the ones that apply Fatigue I to their engulf. but if you're going to play "RAW is the LAW" then I'd recommend the Fatigue wear off much faster than any Fatigue inflicted by other Spirits' engulf, out of balance's sake.
A walkthru, assuming Force 6:
The first thing you do with any attack is to compare AR to DR. For a Force 6 spirit of fire, that's AR13. Note that spirits cannot gain edge, but the defender still potentially can! In this case, the defender must have DR 17+ to score edge from this attack. Also, since the engulf attack only has an AR in the first range band (3 meters), that's the only range it is eligible to be made in.
Next, per pg 223 the Engulf is executed via a Close Combat Attack. A spirit of fire has that skill, and when a spirit has a skill its ranks = Force. Close combat is made with Agility, and in the case of a spirit of fire that's F+2. So when F=6, F+(F+2)=14 dice to make the Engulf attack. If the defender gets more hits the engulf misses, otherwise the engulf hits.
If it hits, as a force 6 it does 6P fire damage. As with most attacks, net hits on the attack raises this DV. Even if the defender soaks all the damage, the power says that the engulf still "happens". Therefore, the engulfed victim always gains the Immobilized status (see pg. 52: this isn't total paralysis... it just means the victim can only make actions that aren't predicated on travelling. So the victim can make close combat attacks against the spirit, but they cannot simply just walk away). Furthermore, the spirit will automatically deal this damage again round by round until its victim escapes.
Now, to the "YMMV" section: how many ranks of Burning should a spirit of fire's engulf inflict? There's no objectively correct answer. However. Note that even without the burning status, the spirit is STILL doing Force damage to the victim every round. If you give Burning X where X=Force, the victim is taking double Force in damage every round. Kind of OP if you ask me. Therefore my preferred way of looking at it is X = number of rounds engulfed. First round, engulf inflicts Force DV. Then Force + 1 DV. Then Force + 2 DV. Up until you're dead, or you escape.