Why can water be a Lewis base?

In this reaction, water is the Lewis base, ${\text{H}}^{+}$ is the Lewis acid (not ${\text{NH}}_{4}^{+}$). Water donates the electron pair, and the ${\text{H}}^{+}$ on ${\text{NH}}_{4}^{+}$ accepts that electron pair to make a bond.
${\text{NH}}_{4}^{+}$ is then a Brønsted-Lowry acid, being a proton donor, and water is then also the Brønsted-Lowry base, being a proton acceptor.