Why can we detect a magnetic field, but not an electric field around an AC current? Consider a wire through which an AC current is passing. Because of the AC current there is an alternating magnetic field present around the wire and because of the alternating magnetic field there exists an electric field perpendicular to the magnetic field, and this produces a propagating EM wave.
If them magnetic field is creating electric field and they combine to form EM waves, why does a compass show a magnetic field around the wire?
My reason for asking this is:
If the electric and magnetic fields combine into a EM wave and we detect the presence of the magnetic field, then why can't we detect the presence of the electric field perpendicular to it?
 A: 
If them magnetic field is creating electric field and they combine to
  form EM waves, why does a compass show a magnetic field around the
  wire?

To produce EM waves, we do need the current to be time varying such that the magnetic field is time varying which induces a time varying electric field etc.
However, it isn't that case that the entire magnetic field produced by the time varying current is associated with EM radiation.
In the reactive near field close to the wire, there are time varying electric and magnetic fields that are not associated with EM waves (which transport energy away) but are, rather, associated with energy storage.  From the linked article:

For example, current flowing in the antenna creates a purely magnetic
  component in the near-field, which then collapses as the antenna
  current begins to reverse, causing transfer of the field's magnetic
  energy back to electrons in the antenna as the changing magnetic field
  causes a self-inductive effect on the antenna that generated it. This
  returns energy to the antenna in a regenerative way, so that it is not
  lost.

This is all quite complicated in general but, for low frequency, e.g., 60Hz AC current along a wire in a circuit, the sinusoidally time varying magnetic field dominates.
Energy is alternately stored in this field (as the current magnitude increases) and returned (as the current magnitude decreases).
