The short answer is that yes you can cast both spells and you get 4 attacks, three of which must be on separate creatures.
The long answer is that I thought I knew the precise answer to this question, but when I researched it I couldn't support my conclusion because there's some ambiguity in the rules.
The current wording of [mwcard=MW1I02]Battle Fury[/mwcard] is
"The next time this round that target creature makes a melee attack, if that attack is not a Counterstrike, at the end of that attack action it may immediately make another quick action melee attack. This extra attack counts as part of the same attack action. Only one Battle Fury spell may be cast per round on the same creature."
The wording of [mwcard=FWI10]Whirling Strike[/mwcard] is unchanged.
The FAQ says this about Battle Fury
The extra attack can be made against the same or a different legal target.
• If the extra attack is made against a different Defender it is treated like a Sweeping Attack, and starts a new attack sequence (with all of the normal steps, including a damage barrier and counterstrike).
• If the extra [attack] is made against the same defender, it will not trigger an additional Counterstrike or Damage Barrier. If the attacking creature becomes affected by a condition acquired from the counterstrike or damage barrier (such as being stunned or dazed), its extra attack will be affected, and the condition marker will remain on the creature until end of the next round (if it has an “end of action phase” effect, as per rules for conditions acquired from a damage barrier or counterstrike).
Since the extra attack is still part of the same attack action (it occurs during the Attack Ends Step of the original attack action), attack bonuses such as Melee +X do not apply to the extra attack.
If you cast Battle Fury on a creature, and the next melee attack that creature makes is a counterstrike attack, Battle Fury will be wasted. (It will not have its effect on a later melee attack that round.)
The effect “Only one Battle Fury spell may be cast per round on the same creature.” is part of the spell’s effect. Thus, if Battle Fury is countered, this effect will not occur, and you would be able to cast a second Battle Fury on this creature this round.
And the FAQ has this to say about Whirling Strike
The additional targets work in the same manner as the Sweeping trait, except there are 2 additional targets instead of 1 additional target. If an attack with additional strikes (such as a “Doublestrike” or “Triplestrike” attack) is made using Whirling Strike, the additional strikes will only be applied to the first target; the subsequent targets will only receive one strike each.
We need to talk a little bit about the difference between an "attack", "attack sequence" and an "attack action." When you use Whirling Strike or an attack from a creature with the Sweeping trait, you make multiple attacks but only one attack action. The "attack sequence" is the steps of the attack 1-8, in order, and your sweeping attack starts a new attack sequence for each target.
On the other hand, with Battle Fury, while you still make all your attacks as part of the same attack action, you only have one attack sequence; the 2nd attack takes place in the final step (step 8) of the initial attack's attack sequence. Presumably, the extra attack has its own steps 1,2,3,4 and 8 that occur within the first attack's step 8.
So!
The easy case is when you chose to attack 4 separate creatures. That's just 4 sweeping attacks.
The complicated case is, as others have noted, using the Battle Fury to attack the same creature twice.
The rules say the Battle Fury extra attack "occurs during the Attack Ends Step of the original attack
action". We have a problem here because the rules seem to be assuming that there will be only one Attack Ends Step per Attack Action. But because we're making multiple attacks (Whirling Attack or Sweeping), we have multiple Attack Ends Steps per attack action.
There's two ways we could resolve this:
1. We could assume that the rules really mean to say that you take the Battle Fury attack at the end of the "original Attack
Sequence" instead of "original Attack
Action." This would mean that you would have to make the Battle Fury's extra attack immediately after the initial attack, without seeing the results of the two extra sweeping attacks.
2. Or, we could take the rules exactly as they're written. The Rules assume something that isn't true, but taking the rules and the cards exactly as written, you don't make the extra attack from Battle Fury until the end of the Attack Action, so you'd make the Battle Fury attack in the final step 8 of the action, after all the sweeping attacks had dealt damage. This would let you make the three sweeping attacks, and then Battle Fury attack any survivor (or a 4th target).
I have no immediate opinion on which of these two is correct. As a playtester I might have rules advice, but I'd have to think about it more.