Is it permissible to deposit a portion of Zakat funds into the account for the construction of the new Children's Cancer Hospital in Egypt, and would that be considered Sadaqah Jariyah (ongoing charity)?

The ruling regarding paying Zakat to charitable projects - such as what you mentioned - has previously been deemed impermissible. Therefore, it is not permissible for you to proceed with that with the intention of fulfilling the obligatory Zakat. However, if you wish to pay it with the intention of charity (sadaqah) and contributing to this great charitable work, we hope that it falls under the ruling of ongoing charity (sadaqah jariyah).