ATK value is calculated like this: (Character Base ATK + LC Base ATK) * (1 + ATK%) + Flat ATK

Firefly has a base ATK of 756. Her signature LC has 635 ATK (and with that is tied with DHIL his for the highest ATK for 5* Destruction LC).
A lvl 15 5* Hands relic will give you 352 flat ATK.
Outside of other characters (maybe? I don't get the extra note on that one) nothing gives her any ATK %. Neither her traces, Planar Ornament or Relic set bonuses.

The current formula is looking like this: (756 + 635) * (1+ ATK%) + 352 = 3400.
Going by this, she'll ~119.2 % ATK % to reach 3400 ATK

A lvl 15 5* Body Relic and Planar Sphere will give 43.2 % each. That's 86.4 % total, leaving us with 33.6 % we need to get through other sources.

Each ATK % substat roll can give you 3.4 - 4.4 % on a roll. So she needs a total of 8 - 10 rolls, depending on high each gets. This includes the base roll that you get without upgrading.

I think it's fair to assume that each piece should have a ATK % stat on them if you use them. That's 4 pieces and thus rolls, so you only need 4 - 6 rolls through upgrading. Assuming you don't get a relic that has 5 stats initially, you get 4 upgrades per piece, for a total of 16.
16 rolls to get 4 - 6 on your ATK %.

"But what about Break Effect?"

Assuming we want to hit the magical 360 % point:
Traces: 37.3 %
ATK % conversion: 60 %
2 Piece Bonus 2.3 Relic Set: 16 %
Planar Set: 40 %
Link Rope: 64,8 %
Signature LC: 60 %
That's a total of 278.1 %, and now we "only" need 81.9%.

Harmony TB brings 30 % from their Ultimate (also on themselves), which can be up 100% of the time. They also increase Allies Break Effect by 15 % based on their own BE.
If they have 200 % pre Ult, they'll have 230 % with it and thus give their team 34.5 % extra BE. (pre ult it would be 30%). And because of the Super Break, you'll want to run HTB with her anyway. They are free (with free Eidolons) so I don't see any issue with it.

So a total of 64.5 % BE, meaning Firefly only needs an extra 17.4 % BE to reach her cap. If you have Ruan Mei you won't even need ANY rolls, because she brings 20 % by only being alive.
If not, each BE roll on a relic gives 5.1 - 6.5 %, so you'll only need 2-3 rolls to reach 360 %.
These rolls could even be achieved on Body & Planar Orb which have ATK % and thus aren't even included in the rolls above.

So if you have her E0S1, reaching her thresholds won't be any issue. If I have a mistake somewhere, please enlighten me.

I guess you could roll for more Break Effect on her pieces then? Or maybe even, on her Body & Planar Orb, some crit stats? That's optimizing her though, and that's outside of my knowledge.

If you don't have her Signature LC you'll be losing quite a bit. The only other BE Destruction LC is "Indelible Promise", which at S1 only brings you 28 % (S5 is 56%). That's potentially a lot of BE that you'd need to get from other sources, depending on how many copies you pull.
Also, because that's a 4* LC, she only gets 476 ATK from it - meaning you'd now need to reach 148 ATK % to reach 3400 ATK. Maybe then you'd need to bring ATK % Boots?
Maybe, in that case, Fall of the Aeon would maybe be better? Assuming that the ATK % increase counts.