Automatic Speech Recognition (ASR) systems often struggle to accurately process children's speech due to its distinct and highly variable acoustic and linguistic characteristics. While recent advancements in self-supervised learning (SSL) models have greatly enhanced the transcription of adult speech, accurately transcribing children's speech remains a significant challenge. This study investigates the effectiveness of layer-wise features extracted from state-of-the-art SSL pre-trained models - specifically, Wav2Vec2, HuBERT, Data2Vec, and WavLM in improving the performance of ASR for children's speech in zero-shot scenarios. A detailed analysis of features extracted from these models was conducted, integrating them into a simplified DNN-based ASR system using the Kaldi toolkit. The analysis identified the most effective layers for enhancing ASR performance on children's speech in a zero-shot scenario, where WSJCAM0 adult speech was used for training and PFSTAR children speech for testing. Experimental results indicated that Layer 22 of the Wav2Vec2 model achieved the lowest Word Error Rate (WER) of 5.15%, representing a 51.64% relative improvement over the direct zero-shot decoding using Wav2Vec2 (WER of 10.65%). Additionally, age group-wise analysis demonstrated consistent performance improvements with increasing age, along with significant gains observed even in younger age groups using the SSL features. Further experiments on the CMU Kids dataset confirmed similar trends, highlighting the generalizability of the proposed approach.