Explanation

‘Insurmountable’ means too great to be overcome, and it collocates naturally with ‘difficulty’ — an insurmountable difficulty is one that cannot be surmounted. ‘Big’, ‘gigantic’ and ‘large’ describe physical size and do not collocate with abstract ‘difficulty’ in standard usage. Since the sentence stresses she faced the difficulty ‘almost’, the adjective must express severity, which only ‘insurmountable’ does. Hence it is the correct choice.