We present the CAlibrating LYMan-$\alpha$ with H$\alpha$ (CALYMHA) pilot survey and new results on Lyman-$\alpha$ (Lya) selected galaxies at z~2. We use a custom-built Lya narrow-band filter at the Isaac Newton Telescope, designed to provide a matched volume coverage to the z=2.23 Ha HiZELS survey. Here we present the first results for the COSMOS and UDS fields. Our survey currently reaches a 3$\sigma$ line flux limit of ~4x10$^{-17}$ erg/s/cm$^{2}$, and a Lya luminosity limit of ~10$^{42.3}$ erg/s. We find 188 Lya emitters over 7.3x10$^5$ Mpc$^{3}$, but also find significant numbers of other line emitting sources corresponding to HeII, CIII] and CIV emission lines. These sources are important contaminants, and we carefully remove them, unlike most previous studies. We find that the Lya luminosity function at z=2.23 is very well described by a Schechter function up to L~10$^{43}$ erg/s with L$^*=10^{42.59+-0.05}$ erg/s, $\phi^*=10^{-3.09+-0.08}$ Mpc$^{-3}$ and $\alpha$=-1.75+-0.15. Above L~10$^{43}$ erg/s the Lya luminosity function becomes power-law like, driven by X-ray AGN. We find that Lya-selected emitters have a high escape fraction of 37+-7%, anti-correlated with Lya luminosity and correlated with Lya equivalent width. Lya emitters have ubiquitous large (~40 kpc) Lya haloes, 2x larger than their Ha extents. By directly comparing our Lya and Ha luminosity functions we find that the global/overall escape fraction of Lya photons (within a 13 kpc radius) from the full population of star-forming galaxies is 5.1+-0.2% at the peak of the star formation history. An extra 3.3+-0.3% of Lya photons likely still escape, but at larger radii.