How can I find the bandwidth of a sinc function?
It depends on the definition of the \$\mathrm{rect}\$ function you are using, but the most likely confusion is \$\omega=2\pi{}f\$, not \$\pi{}f\$.

However, in the usual definition of \$\mathrm{rect}\$, for example as given by Wikipedia,  the bandwidth of \$\mathrm{rect}(\frac{\omega}{20,000\pi})\$ is normally stated as 5 kHz, not 10 kHz, because we consider only the portion of the passband in the positive frequencies.