The statement "insulated bucket" tells one that there is no heat transfer between the ice and water and the outside world.
This means that the overall average number of hydrogen bonds made between the water molecules cannot change as it takes energy to break a bond.
The statement that both the ice and the water are at the same temperature means that if they come together they will be in thermal equilibrium which means that there will be no net heat transfer between them. On the molecular scale this means that on average when liquid water molecules collide with ice molecules there is no net transfer of kinetic energy between them.
So you have ice with each water molecule having four hydrogen bonds and liquid with on average each water molecule having slightly less than four hydrogen bonds coexisting together.
There can be an interchange as to which hydrogen bonds are actually made or broken but overall the number stays the same.
The interesting thing is that you allowed some heat into the bucket then as long as there was ice present the temperature would stay at $0^\circ \rm C$ with some of the ice turning into water.