We present the discovery of a giant 100 kpc Lyα nebula detected in the core of the X-ray emitting cluster CL J1449+0856 at z = 1.99 through Keck/LRIS narrow-band imaging. This detection extends the known relation between Lyα nebulae and overdense regions of the universe to the dense core of a 5-7 × 1013 M o cluster. The most plausible candidates to power the nebula are two Chandra-detected AGN host cluster members, while cooling from the X-ray phase and cosmological cold flows are disfavored primarily because of the high Lyα to X-ray luminosity ratio (LLyα/Lx ≈ 0.3, ≳ 10-1000 times higher than in local cool-core clusters) and by current modeling. Given the physical conditions of the Lyα-emitting gas and the possible interplay with the X-ray phase, we argue that the Lyα nebula would be short-lived (≲10 Myr) if not continuously replenished with cold gas at a rate of 1000 M o yr-1. We investigate the possibility that cluster galaxies supply the required gas through outflows and we show that their total mass outflow rate matches the replenishment necessary to sustain the nebula. This scenario directly implies the extraction of energy from galaxies and its deposition in the surrounding intracluster medium (ICM), as required to explain the thermodynamic properties of local clusters. We estimate an energy injection of the order of per particle in the ICM over a 2 Gyr interval. In our baseline calculation, AGNs provide up to 85% of the injected energy and two-thirds of the mass, while the rest is supplied by supernovae-driven winds.