Firstly:
yet a norm implies an inner product -- which is to say a notion of length implies a notion of angle?
A norm actually need not specify an inner product. There are norms which do not come from an inner product.
Let's be more specific. For $\newcommand{\nc}{\newcommand}
\nc{\para}[1]{\left( #1 \right)}
\nc{\abs}[1]{\left| #1 \right|}
\nc{\br}[1]{\left[ #1 \right]}
\nc{\set}[1]{\left\{ #1 \right\}}
\nc{\ip}[1]{\left \langle #1 \right \rangle}
\nc{\n}[1]{\left\| #1 \right\|}
\nc{\norm}[1]{\left\| #1 \right\|}
\nc{\floor}[1]{\left \lfloor #1 \right \rfloor}
\nc{\ceil}[1]{\left \lceil #1 \right \rceil}
\nc{\setb}[2]{\set{#1 \, \middle| \, #2}}
\nc{\dd}{\mathrm{d}}
\nc{\dv}[2]{\frac{\dd #1}{\dd #2}}
\nc{\p}{\partial}
\nc{\pdv}[2]{\frac{\partial #1}{\partial #2}}
\nc{\a}{\alpha}
\nc{\b}{\beta}
\nc{\g}{\gamma}
\nc{\d}{\delta}
\nc{\ve}{\varepsilon}
\nc{\t}{\theta}
\nc{\m}[1]{\begin{bmatrix} #1 \end{bmatrix}}
\nc{\C}{\mathbb{C}}
\nc{\N}{\mathbb{N}}
\nc{\R}{\mathbb{R}}
\nc{\P}{\mathbb{P}}
\nc{\Q}{\mathbb{Q}}
\nc{\Z}{\mathbb{Z}}
\nc{\AA}{\mathcal{A}}
\nc{\BB}{\mathcal{B}}
\nc{\CC}{\mathcal{C}}
\nc{\FF}{\mathcal{F}}
\nc{\GG}{\mathcal{G}}
\nc{\II}{\mathcal{I}}
\nc{\JJ}{\mathcal{J}}
\nc{\KK}{\mathcal{K}}
\nc{\PP}{\mathcal{P}}
\nc{\RR}{\mathcal{R}}
\nc{\SS}{\mathcal{S}}
\nc{\TT}{\mathcal{T}}
\nc{\UU}{\mathcal{U}}
V$ a vector space over a field $F$, an inner product $\ip{\cdot,\cdot}$ and a norm $\n{\cdot}$ on $V$ are functions that meet certain properties. An inner product is special in this, given one, it defines a norm:
$$
\norm{x} := \sqrt{\ip{x,x}}
$$
However, a norm need not define an inner product. (After all, one needs to take in two vectors, and the other just one.) One can show, for instance, that a norm is induced by an inner product if and only if the norm satisfies the parallelogram law (MSE post):
$$2 \n{x}^2 + 2 \n{y}^2 = \n{x-y}^2 + \n{x+y}^2$$
Examples would be the so-called $p$-norms; when $p\ne 2$, they are not induced by an inner product. Recall that we define, for $x := (x_i)_{i=1}^n \in \R^n$,
$$\n{x}_p := \para{ \sum_{i=1}^n \abs{x_i}^p }^{1/p}$$
(Note that our familiar Euclidean norm is the $p=2$ norm, and is induced by the dot product.)
Now onto your question: essentially, how do inner products and angles relate?
In $\R^n$, under the usual scenarios (Euclidean distance and norm, inner product is the dot product), we may define the angle $\t$ between $x,y \in \R^n$ by
$$\t = \arccos \para{ \frac{\ip{x,y}}{\n{x}\n{y}}}$$
This comes from one way of defining the dot product:
$$\ip{x,y} := \n{x} \n{y} \cos \t$$
This $\t$ matches up with the angle we think of in the ordinary sense. We can see why the $\t$ arises in the following way...
First, take it as given that we define
$$
\ip{x,y} := \sum_{i=1}^n x_i y_i
$$
One can prove a polarization identity of inner products:
$$\ip{x,y} = \frac{\n{x+y}^2 - \n{x-y}^2}{4}$$
One also has the law of cosines. In the language of vectors, one has that
$$\n{x-y}^2 = \n{x}^2 + \n{y}^2 - 2 \n{x} \n{y} \cos \t$$
for $\t$ (in the geometric sense) the angle between $x,y$. But, using that this norm $\n \cdot$ is induced by $\ip{\cdot,\cdot}$, and various properties of inner products in general, one has that
$$\n{x-y}^2 = \n{x}^2 + \n{y}^2 - 2 \ip{x,y}$$
Equating these two thus yields
$$\ip{x,y} = \n{x} \n{y} \cos \t$$
Of course, looking at an inner product in general, how much do we really know? I tell you that $\ip{x,y} = 0.35$; does this tell us anything?
It does tell us one key property of very common interest all throughout mathematics -- that the vectors are not orthogonal. Two vectors are orthogonal if and only if $\ip{x,y} = 0$.
In the Euclidean-$\R^n$ sense, this amounts to meeting at right angles in the plane they span. Of course, we have long-since generalized this notion to other spaces, e.g. functions, on which the axioms of an inner product can be met, even if the notion of "angle" becomes fuzzy, because orthogonality makes it very easy to represent elements of a vector space in certain bases (bases of elements which are pairwise orthogonal).
Much of the elegance and applicability of Fourier analysis, for instance, comes from the fact that $\set{\sin(kx),\cos(kx)}_{k=1}^\infty$ forms an orthogonal basis of square-integrable functions under the inner product
$$\ip{f,g}_{L^2[-\pi,\pi]} := \int_{-\pi}^\pi f(x) g(x) \, \dd x$$
In particular,
$$\begin{align*}
\int_{-\pi}^\pi \sin(mx) \cos(nx) \, \dd x &= 0 \\
\int_{-\pi}^\pi \sin(mx) \sin(nx) \, \dd x &= \begin{cases}
\pi, & m = n \\ 0 , & \text{otherwise} \end{cases} \\
\int_{-\pi}^\pi \cos(mx) \cos(nx) \, \dd x &= \begin{cases}
\pi, & m = n \\ 0 , & \text{otherwise} \end{cases}
\end{align*}$$
In fact, "nice enough" functions can be easily written as an infinite sum of (scaled and modulated) sines and cosines owing to this fact: a Fourier series.
