Solution:

Let $K = \frac{n(n+1)}{2},$ so the sum of cubes is $K^2.$ Modulo $n + 5$ we have $n \equiv -5,$ hence $2K = n(n+1) \equiv (-5)(-4) = 20$ and $4K^2 \equiv 400.$ If $K^2 \equiv 17 \pmod{n+5},$ then $n + 5$ divides $400 - 4 \cdot 17 = 332 = 2^2 \cdot 83.$ Since a remainder of $17$ also forces $n + 5 \gt 17,$ the candidates are $n + 5 \in \{83, 166, 332\},$ i.e. $n \in \{78, 161, 327\}.$

Because we multiplied by $4,$ each candidate must be checked. For $n = 78:$ $K = 3081 \equiv 10 \pmod{83},$ and $10^2 = 100 \equiv 17 \pmod{83}.$ For $n = 161:$ $K = 13041 \equiv 93 \pmod{166},$ and $93^2 = 8649 = 52 \cdot 166 + 17.$ For $n = 327:$ $K = 53628 \equiv 176 \pmod{332},$ and $176^2 = 30976 = 93 \cdot 332 + 100,$ remainder $100 \ne 17,$ so this one fails.

The valid values are $n = 78$ and $n = 161,$ with sum $78 + 161 = 239.$