We propose time-frequency domain methods for noise estimation and speech enhancement. A speech presence detection method is used to find connected time-frequency regions of speech presence. These regions are used by a noise estimation method and both the speech presence decisions and the noise estimate are used in the speech enhancement method. Different attenuation rules are applied to regions with and without speech presence to achieve enhanced speech with natural sounding attenuated background noise. The proposed speech enhancement method has a computational complexity, which makes it feasible for application in hearing aids. An informal listening test shows that the proposed speech enhancement method has significantly higher mean opinion scores than minimum mean-square error log-spectral amplitude (MMSE-LSA) and decision-directed MMSE-LSA.