Let me focus on math part only and explain how your expected life rate depends on the probability of a death in a given year. I will use your assumption that this probability of death does not change so let's assume it is $p$. $p$ has to be greater than $0$ and less than $1$ (that's a general requirement for probabilistic). We'll get back to this value later.
So $p$ is your probability of dying during a specific year if you were alive in the beginning of it and similarly the remaining $1 - p$ is the probability you survive another year.
Now to die at the age of $n$ you need to live for $n - 1$ years and then die in the last, $n$-th year. which means a probability of such event is
$$
P_n = ( 1 - p )^{n-1} * p
$$
An expected lifespan is an expected value of a random variable of your age with a probability to reach a value $n$ as $P_n$ calculated above. In other words it is
$$
E=\sum_{n=1}^\inf (n*P_n)
$$
Lets put the $P_n$ values into the equation
$$
E = \sum_{n=1}^\inf ( n * p * (1-p) ^ {n-1} )
$$
Let's extract constant so that the power is re-indexed to n
$$
E = \frac{p}{1-p} * \sum_{n=1}^\inf (n * (1-p)^n)
$$
Since for $n=0$ the value of $n*(1-p)^n$ is also $0$ (since we multiply everything by $n$ which is $0$ so we can reindex the whole sum to start from $0$
$$
E = \frac{p}{1-p} * \sum_{n=0}^\inf (n * (1-p)^n)
$$
Now let's apply a formula to calculate this infinite sum and calculate it and as a result we get
$$
E = \frac{p}{1-p} * \frac{1-p}{(1 - (1-p))^2} = \frac{p}{p^2} = \frac{1}{p}
$$
In other words your expected lifespan is exactly $1/p$ where $p$ is probability of death during one year.
Now let's apply various values provided in other answers and comments:
- MichaelK's comment gives $p=0.00181$ which accounts to the average lifespan of $552$ years (based on statistics from 2000)
- Dayton Williams in his answer gives $p=\frac{1}{1576}$ which accounts to the average lifespan of $1576$ years
- Chris Becke's answer gives $p$ between $0.0003$ and $0.0006$ which accounts to the average lifespan between $1667$ and $3333$ years (various statistics, 2010 - 2012)
etc.
Now depending on details you can modify what you include or exclude into your death cause that can kill an ageless person and adapt values accordingly.
This is a pure mathematical approach ignoring few things though:
- The exposure to dangers of external world are reduced until you are close to being mature. You might add 15 years (or something like that) to your results to cover this aspect
- The probability of death change over time (ignored as explicitly requested by OP, but in general it decreases over time except a situations like war so it might increase the average lifespan)
- The ageless person gathers experience that should help them survive even longer, e.g. notice the warning signs and avoid most dangerous situations resulting in a lower death probability than a normal mortal
- Ageless person may attract various people due to their lengthy life. It might increase their death probability and as a result reduce the lifespan
- There might be other aspects impacting the lifespan of an ageless person that are not influencing normal mortals and are hard to predict (you have an area to add whatever you think relevant changing this average according to your needs)
