It's a metaphor that implies skill, meticulousness and balance. Possibly feet - more on that with the examples.
In your first example you have a person who is providing health care to more children than they can reasonably care for. He's doing his best to help the sickest children, while keeping the others stable. These are conflicting priorities and balance is crucial. The sick children are in themselves delicate, and he must be meticulous and skillful in his care to avoid harming them further. The metaphor of a dance in this case implies a doctor or nurse that is moving briskly on his feet around a hospital (or similar).
In your second example, there's a "dance partner". The committee leaders must carefully interact with Mueller, balancing their own desired outcomes against Mueller's own motivations and duties. They must handle the negotiations with care in case a small mistake causes a complete breakdown of communication. This one has a couple of foot-related idioms that relate to it. They might not want to put a foot wrong (make a conspicuous mistake) or step on anyone's toes (offend someone, in particular by interfering with their responsibilities). They might also be described as walking on egg shells - a more obvious metaphor where stepping carefully to avoid breaking something fragile is used to illustrate the need to proceed slowly and with great care and caution.
Hopefully that helps!
