Not sure you can do a traditional with inherited cash since technically the taxes have already been paid on that amount. I would probably put it into a Roth IRA (after tax) and you can contribute 11K ($5,500 for 2013, same for 2014) if you do the contributions before April to get the 2013 contribution.
In my experience, you don't pay taxes on a cash inheritance; that is paid through the estate if there is anything owed, and if you were left a sum of money, then it is yours and would have already had taxes taken out. Of course, I have only my experience to go by, so you'd need to double check through the IRS or the like, but we inherited the sum of $25K through a family friend that left that to my husband a while back, and the IRS said that no taxes were owed by us at the time (it's not income either). And I inherited a substantial amount of cash money from my dad recently, and same thing again. If it is inherited in an account like an IRA, then there could be taxes needed to be taken out if you want to access the funds, but you can contact the brokerage holding the account for details on that aspect.
Not sure why you say you'd need to buy index funds monthly - you can just open the Roth and put in the money, and buy index funds all right then, and with the amount over 10K, you can probably get the admiral shares since that's usually the minimum investment amount.
Tax wise, you'd probably do better at increasing your 401K contributions to lower your overall income if you're right on the cusp of being in one or the other tax bracket.