Invulnerability/Logia
Physical Attacks
While utilizing Shikigami no mae, Konan is a logia which means that physical attacks are useless against her. If she can casually split herself into thousands of pieces, it means that cutting her is useless as she can function just fine even if an insignificant number of papers get cut. Blunt damage is also not going to be effective because
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and blunt damage doesn't usually cause paper to get obliterated due to their flexibility.
Hot Attacks
As you would expect, katon is a weakness of paper considering the paper covering Ryusui's body disappeared after defending him from the katon. (
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)(
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). Explosions can still kill Konan considering they can vaporize her body if strong enough to do so considering
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explosion. From this, we can agree that her logia abilities don't help against heat attacks.
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Lightning Attacks
Generally speaking, lightning attacks are useless because electrocution is useless against someone without a blood stream but extremely powerful lightning attacks should produce enough heat to vaporize parts of her body if not all, but those are rare. One might argue that raiton would cause paper to get crumpled up but crumpled up paper can still be manipulated as Konan herself has already done so.
All in all, Konan is very hard to put down but wide scale attacks which can obliterate her can kill her as well as attacks that absorb chakra.