Goshikuro feeds off of negative energy, but at this point neither Kurama, nor Yoko, fully realizes that these demons are just the subordinates of another, more powerful entity they faced in the past. So by knowing that Botan was attacked and placed in danger, Kurama is unknowingly exuding a dangerous amount of anger that would in turn strengthen Goshikuro. I think sadness contributes a lot to his recovery, but the rage of someone like Kurama is double that, so that was why Botan was moved on. I don't know the true level of how it will work, but let's say Kurama being angered for even a moment is enough to put Goshikuro's recovery a week ahead, etc etc. It's a work in progress.
I'm thinking Kurama enters his Yoko form and overwhelms the invisible hold of demon energy within Botan--after resuscitating her, of course-- but that does leave the demon that put her at risk in the first place. I think that's a plan they decide on once he makes sure Botan is freed and safe. The demon will have exposed themselves to Kurama, but it could also probably change bodies, and I think only in Yoko form Kurama is sensitive enough to demon energy to sniff them out. Thoughts?