In your textbook, it says that given a certain money supply, the interest rate must increase if the demand for money increases.
The money supply is determined by the central bank, which can buy bonds (which takes bonds out of circulation and increases the supply of money in circulation), or sell bonds (putting bonds in circulation but decreasing the supply of money in circulation). It can also, as in your example, do nothing.
If the demand for money increases, then market participants are willing to either "buy" more money at a given price (where the "price" of money is the interest rate), or pay a higher price for a given amount of money. This is true of all goods— and so long as demand isn't either perfectly elastic or perfectly inelastic, both are true.
So since in your example the money supply is fixed (i.e., the supply curve is perfectly inelastic), if the demand for money increases, the interest rate will increase. The interest rate increasing as a result of increased demand for money isn't dependent on the central bank doing anything, it's dependent on the central bank doing nothing.
In practice, central banks usually target a particular interest rate. When they do so, they can't directly observe the demand for money, but they can observe the equilibrium price (i.e., the interest rate). So what they do is engage in open-market operations— buying [or selling] bonds to increase [or decrease] the supply of money, in turn decreasing [or increasing] the interest rate until it is close to their target.