Why don't I get an implied volatility skew?

I am trying to implement some financial formulas in Excel, and I sadly do not see an implied volatility smile, as I am being told I should. What's going on here?

If rate is zero, spot is 100, maturity is 1 year, the call price is 10, and strike ranges from 80 to 150, then I get a rising implied volatility when strike goes from 90 to 150, and for strike equal to 80 or thereabouts, I get an implied volatility of 0.999.

Implied volatility is the volatility implied by plugging market prices and other observable variables into the Black-Scholes formula and solving for volatility of the underlying that would lead to that price. I see no evidence in your post that you are using market prices. If you use made-up prices, you can solve for implied volatility but there is no reason to think you will see a volatility smile.

In other words, the volatility smile is not a result of the Black-Scholes formula or standard options pricing theory, but a breakdown of it. If the assumptions of Black-Scholes were satisfied, then options prices would be such that there would be no volatility smile whatsoever. We know that Black-Scholes theory is not exactly right because far out of the money options cost a little more than Black-Scholes theory predicts.

TL;DR answer: the volatility smile is an empirical observation about market prices. Because you are not using market prices, you should not expect to see a smile.

If rate is zero, spot is 100, maturity is 1 year, the call price is 10, and strike ranges from 80 to 150, then I get a rising implied volatility when strike goes from 90 to 150

You are assuming that call prices are constant when the strike changes, which is not true. The price of a call decreases as strike increases, and rises as volatility rises, so if you decrease the strike but leave the call price alone, the implied volatility will be higher to compensate.

If instead of a constant call price, you look at actual market prices of calls at various strikes, you should see a "smile" of implied vols, with the at-the-money vol being the lowest.

Note that this does not always happen, but is a very common result of actual market prices.