since 1 l of liquid water equals 1 kg, you can easily convert your kg figures into l figures. Then convert s into month, so that you end up with a figure giving the amount of rainfall in l per m-2 per month. Luckily, l / m-2 equals the unit of mm of precipitation, so that you have already arrived at your desired figure of mm of precipitation/month :-)
days= mm/month =P [kg m-2 s-1] * 86400 * N days of a given month= 86400 PN mm/month. For example, you have a rainfall magnitude of P [kg m-2 s-1] in the month of January. Then, you just need to multiply that magnitude P [kg m-2 s-1] with 86400 * 31 days in January. Therefore, the rainfall in mm/month = P [kg m-2 s-1] * 86400 * 31.