We have mapped the dense dark core L1544 in H¹³CO⁺ (1-0), DCO⁺ (2-1), DCO⁺ (3-2), N₂H⁺ (1-0), N₂H⁺ (3-2), N₂D⁺ (2-1), N₂D⁺ (3-2), C¹⁸O (1-0), and C¹⁷O (1-0) using the IRAM 30 m telescope. We have obtained supplementary observations of HC¹⁸O⁺ (1-0), HC¹⁷O⁺ (1-0), and D¹³CO⁺ (2-1). Many of the observed maps show a general correlation with the distribution of dust continuum emission, in contrast to C¹⁸O (1-0) and C¹⁷O (1-0), which give clear evidence for depletion of CO at positions close to the continuum peak. In particular, N₂D⁺ (2-1) and (3-2) and to a lesser extent N₂H⁺ (1-0) appear to be excellent tracers of the dust continuum. Our DCO⁺ maps have the same general morphology as the continuum while H¹³CO⁺ (1-0) is more extended. We find also that many apparently optically thin spectral lines such as HC¹⁸O⁺ and D¹³CO⁺ have double or highly asymmetric profiles toward the dust continuum peak. We have studied the velocity field in the high-density nucleus of L1544, putting particular stress on tracers such as N₂H⁺ and N₂D⁺, which trace the dust emission and which we therefore believe trace the gas with density of order 10⁵ cm^-3. We find that the tracers of high-density gas (in particular, N₂D⁺) show a velocity gradient along the minor axis of the L1544 core and that there is evidence for larger line widths close to the dust emission peak. We interpret this using the model of the L1544 proposed by Ciolek and Basu and by comparing the observed velocities with those expected on the basis of their model. The results show reasonable agreement between observations and model in that the velocity gradient along the minor axis and the line broadening toward the center of L1544 are predicted by the model. This is evidence in favor of the idea that ambipolar diffusion across field lines is one of the basic processes leading to gravitational collapse. However, the double-peaked nature of the profiles is reproduced only at the core center and if a "hole" in the molecular emission, due to depletion, is present. Moreover, line widths are significantly narrower than observed and are better reproduced by the Myers & Zweibel model, which considers the quasi-static vertical contraction of a layer due to dissipation of its Alfvénic turbulence, indicating the importance of this process for cores on the verge of forming a star.