I don't know what you mean by "I instructed this distribution to go straight to my IRA." or why the IRA custodian of your inherited IRA accepted this instruction or your
own IRA custodian accepted the money without any accompanying paperwork.
Distributions from an IRA cannot be rolled over into another IRA. There is
an exception to this statement where the distribution is a Rollover from one of your own IRA accounts that you plan on taking in cash and rolling over into your another of your IRA accounts (with a different custodian) within the next 60 days. As a precaution, the IRS insists that the custodian making the distribution withhold 20% of the amount as Federal income tax and you have to make up the difference and send the full amount within the 60-day grace period to the new custodian. If you miss the
deadline for any reason whatsoever, you are deemed to have received a distribution
from your own IRA, have to pay tax on it, and possibly
a penalty for early withdrawal too. The money sent in late to the new custodian
is also a problem because it is likely an excess contribution to your
IRA. So you can take a short-term loan from your
IRA in a sense but you do have to make the US Government a loan of the 20% withholding until you get it back as a tax refund, and you have to make up the 20% withheld
and send the full amount to the new custodian. So most people find it more convenient
to arrange a custodian-to-custodian transfer where there is no withholding and no
60-day grace period to contend with.
That being said, if you are getting a distribution from an inherited IRA, that
distribution cannot be rolled over into your own IRA. But, money is fungible,
and if what happened is that you got a check from the custodian of the Inherited
IRA and endorsed it to be payable to the custodian of your own IRA, then you
have made a contribution to your own IRA for that year. If the amount of
the check exceeded your maximum allowable contribution
for the year under consideration, you have made an excess contribution and
are liable for quite severe monetary penalties. Withdraw the money ASAP,
and remember that you are no longer to make additional contributions for
that year.
Finally, as I noted in an answer in
another thread, you have to
have earned income
(W2 wages or self-employment income reported on Schedule C) in order to
make an IRA contribution, so be sure to check that matter too.