This confirms what I said before, that you received a distribution that was not eligible for rollover despite the instructions that you gave the old IRA custodian to perform a nonreportable trustee-to-trustee transfer. The tax code has no provision make the deposit of these funds into the new inherited IRA permissible because the funds came under your control when the check was made out to you and you cashed it. In fact, the new IRA custodian should not have allowed these funds to be deposited into the new inherited IRA knowing that the funds had been paid to you and not directly to them for your benefit. This means that the deposit into the new IRA constitutes an excess contribution (and the new IRA fails to qualify as an inherited IRA despite its titling).
It seems that your only recourse would be legal action against the old custodian in an attempt to obtain compensation you for whatever losses will result from not being able to spread out the distributions out over multiple years, but that could be a lost cause if the old IRA custodian is no longer in business. You might want to talk to an attorney experienced in such matters.
It seems unlikely that you would have any success attempting to claim that the portion deposited into the new inherited IRA was actually done by nonreportable trustee to trustee transfer because you won't be able to show that the funds were paid to the new IRA custodian and not to you.
Mistaken distributions from inherited IRAs when a non-spouse beneficiary attempts to move the IRA have been problems often enough for people that Congress has considered changing the tax code to allow such a distribution to be rolled over, but that change has not been made. Such distributions are not permitted to be deposited back into an inherited IRA.
