We use three-dimensional simulations to study injection and electron beam quality in laser wakefield acceleration (LWFA) using the density transition technique. We vary the density transition length scale, covering both the sharp and gradual density transition regimes. We find that the injected charge decreases monotonically as the density transition scale length increases, and as a consequence the energy-spread and emittance improve monotonically. Therefore, there is no optimal transition length that gives the best quality beam, contrary to earlier suggestions. However, the density transition technique does give high brightness electron beams with kA current, energy-spread of around 1% and normalized rms emittance of around 1π mm-mrad. We study the application of these LWFA beams as drivers for a short-wavelength free-electron laser (FEL), using analytic formulae as well as three-dimensional simulations. Because higher current favours a shorter transition length, while smaller energy-spread and emittance favour a longer transition length, there is now an optimal density transition scale length (for our parameters, 50 µm) that gives the best FEL performance: lasing at 270 nm, with a saturated power of around 360 MW, over an undulator length of only 6 m. Further improvements, like lower plasma density and laser guiding, could result in GeV-class beams of sufficient brightness to drive a soft x-ray FEL.