Spectroscopy with the James Webb Space Telescope has opened the possibility of identifying moderate-luminosity active galactic nuclei (AGNs) in the early Universe, at and beyond the epoch of re-ionisation, complementing previous surveys of much more luminous (and much rarer) quasars. We present 12 new AGNs at 4$<$z$<$7 in the JADES survey (in addition to the previously identified AGN in GN-z11 at z=10.6) revealed through the detection of a broad-line region (BLR) seen in the Balmer emission lines. The depth of JADES, together with the use of three different spectral resolutions, enables us to probe a lower-mass regime relative to previous studies. In a few cases, we find evidence for two broad components of Halpha , which suggests that these could be candidate merging black holes (BHs), although a complex BLR geometry cannot be excluded. The inferred BH masses range from $ 10^7 M_ down to $ 10^5 M_ interestingly probing the regime expected for direct collapse BHs. The inferred AGN bolometric luminosities ($ $ erg/s) imply accretion rates that are $< 0.5$ times the Eddington rate in most cases. However, small BHs, with $ BH 10^6 M_ tend to accrete at Eddington or super-Eddington rates. These BHs at zsim 4-11 are over-massive relative to their host galaxies' stellar masses when compared to the local $ BH -M_ star $ relation, even approaching $ BH star $, as was expected from heavy BH seeds and/or super-Eddington accretion scenarios. However, we find that these early BHs tend to be more consistent with the local relation between $ BH $ and velocity dispersion, as well as between $ BH $ and dynamical mass, suggesting that these are more fundamental and universal relations. On the classical, optical narrow-line excitation-diagnostic diagrams, these AGNs are located in the region that is that is locally occupied by star-forming galaxies, implying that they would be missed by the standard classification techniques if they did not display broad lines. Their location on the diagram is consistent with what is expected for AGNs hosted in metal-poor galaxies ($ Z 0.1-0.2 Z_ The fraction of broad-line AGNs with $ AGN erg/s$ among galaxies in the redshift range of $4<z<6$ is about 10<!PCT!>, suggesting that the contribution of AGNs and their hosts to the re-ionisation of the Universe is $>$10<!PCT!>.