  We present an analytical model that considers energy arising from a magnetar
central engine. The results of fitting this model to the optical and X-ray
light curves (LCs) of five long-duration $\gamma$-ray bursts (LGRBs) and two
ultra-long GRBs (ULGRBs), including their associated supernovae (SNe), show
that emission from a magnetar central engine cannot be solely responsible for
powering an LGRB-SN. While the early AG-dominated phase can be well described
with our model, the predicted SN luminosity is underluminous by a factor of
$3-17$. We use this as compelling evidence that additional sources of heating
must be present to power an LGRB-SN, which we argue must be radioactive
heating. Our self-consistent modelling approach was able to successfully
describe all phases of ULGRB 111209A / SN 2011kl, from the early afterglow to
the later SN, where we determined for the magnetar central engine a magnetic
field strength of $1.1-1.3\times10^{15}$ G, an initial spin period of
$11.5-13.0$ ms, a spin-down time of $4.8-6.5$ d, and an initial energy of
$1.2-1.6\times10^{50}$ erg. These values are entirely consistent with those
determined by other authors. The luminosity of a magnetar-powered SN is
directly related to how long the central engine is active, where central
engines with longer durations give rise to brighter SNe. The spin-down
timescales of superluminous supernovae (SLSNe) are of order months to years,
which provides a natural explanation as to why SN 2011kl was less luminous than
SLSNe that are also powered by emission from magnetar central engines.
